From 71a0c7ee41858f83ffb2b81524b4b3c1ca1864be Mon Sep 17 00:00:00 2001
From: Thomas Dehaeze <dehaeze.thomas@gmail.com>
Date: Wed, 2 Dec 2020 11:00:11 +0100
Subject: [PATCH] Rework introduction

---
 figs/general_conf_shaping_S_GS.pdf           |  Bin 0 -> 72942 bytes
 figs/general_conf_shaping_S_GS.png           |  Bin 0 -> 18841 bytes
 figs/general_conf_shaping_S_GS.svg           |  228 +++
 figs/general_conf_shaping_S_T_GS.pdf         |  Bin 0 -> 73336 bytes
 figs/general_conf_shaping_S_T_GS.png         |  Bin 0 -> 21860 bytes
 figs/general_conf_shaping_S_T_GS.svg         |  261 +++
 figs/mixed_sensitivity_control_schematic.pdf |  Bin 0 -> 73343 bytes
 figs/mixed_sensitivity_control_schematic.png |  Bin 0 -> 21938 bytes
 figs/mixed_sensitivity_control_schematic.svg |  255 +++
 figs/mixed_sensitivity_ref_tracking.pdf      |  Bin 73687 -> 54484 bytes
 figs/mixed_sensitivity_ref_tracking.png      |  Bin 19499 -> 16806 bytes
 figs/mixed_sensitivity_ref_tracking.svg      |  218 ++-
 index.html                                   | 1066 ++++++++----
 index.org                                    | 1587 +++++++++++++++++-
 14 files changed, 3124 insertions(+), 491 deletions(-)
 create mode 100644 figs/general_conf_shaping_S_GS.pdf
 create mode 100644 figs/general_conf_shaping_S_GS.png
 create mode 100644 figs/general_conf_shaping_S_GS.svg
 create mode 100644 figs/general_conf_shaping_S_T_GS.pdf
 create mode 100644 figs/general_conf_shaping_S_T_GS.png
 create mode 100644 figs/general_conf_shaping_S_T_GS.svg
 create mode 100644 figs/mixed_sensitivity_control_schematic.pdf
 create mode 100644 figs/mixed_sensitivity_control_schematic.png
 create mode 100644 figs/mixed_sensitivity_control_schematic.svg

diff --git a/figs/general_conf_shaping_S_GS.pdf b/figs/general_conf_shaping_S_GS.pdf
new file mode 100644
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diff --git a/figs/general_conf_shaping_S_T_GS.pdf b/figs/general_conf_shaping_S_T_GS.pdf
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diff --git a/figs/mixed_sensitivity_ref_tracking.pdf b/figs/mixed_sensitivity_ref_tracking.pdf
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diff --git a/figs/mixed_sensitivity_ref_tracking.png b/figs/mixed_sensitivity_ref_tracking.png
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diff --git a/index.html b/index.html
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@@ -3,7 +3,7 @@
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 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-12-01 mar. 18:49 -->
+<!-- 2020-12-02 mer. 11:00 -->
 <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" />
@@ -30,95 +30,111 @@
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#org86f37ed">1. Introduction to Model Based Control</a>
+<li><a href="#org267ac69">1. Introduction to Model Based Control</a>
 <ul>
-<li><a href="#org3c879ae">1.1. Model Based Control - Methodology</a></li>
-<li><a href="#org3930ba3">1.2. From Classical Control to Robust Control</a></li>
-<li><a href="#orgc80bc10">1.3. Example System</a></li>
+<li><a href="#org2816731">1.1. Model Based Control - Methodology</a></li>
+<li><a href="#org18d6909">1.2. From Classical Control to Robust Control</a></li>
+<li><a href="#org4fde82b">1.3. Example System</a></li>
 </ul>
 </li>
-<li><a href="#org5e5539a">2. Classical Open Loop Shaping</a>
+<li><a href="#org1bfaa00">2. Classical Open Loop Shaping</a>
 <ul>
-<li><a href="#orgacb6d03">2.1. Introduction to Loop Shaping</a></li>
-<li><a href="#orgfac68ae">2.2. Example of Open Loop Shaping</a></li>
-<li><a href="#orgcc18a4a">2.3. \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
-<li><a href="#orgc6b8e73">2.4. Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
+<li><a href="#org4d52f02">2.1. Introduction to Loop Shaping</a></li>
+<li><a href="#orgfefb3c0">2.2. Example of Open Loop Shaping</a></li>
+<li><a href="#org6fdc1bc">2.3. \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
+<li><a href="#org1172944">2.4. Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
 </ul>
 </li>
-<li><a href="#org397a600">3. A first Step into the \(\mathcal{H}_\infty\) world</a>
+<li><a href="#orgde10b7a">3. A first Step into the \(\mathcal{H}_\infty\) world</a>
 <ul>
-<li><a href="#org60d66b0">3.1. The \(\mathcal{H}_\infty\) Norm</a></li>
-<li><a href="#org66d736a">3.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
-<li><a href="#org7054eb2">3.3. The Generalized Plant</a></li>
-<li><a href="#orga725c9e">3.4. The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant</a></li>
-<li><a href="#orgfec6fc7">3.5. From a Classical Feedback Architecture to a Generalized Plant</a></li>
+<li><a href="#org5ae8918">3.1. The \(\mathcal{H}_\infty\) Norm</a></li>
+<li><a href="#org91b1ae9">3.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
+<li><a href="#org9c4b033">3.3. The Generalized Plant</a></li>
+<li><a href="#org0187c7b">3.4. The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant</a></li>
+<li><a href="#org822d770">3.5. From a Classical Feedback Architecture to a Generalized Plant</a></li>
 </ul>
 </li>
-<li><a href="#orgb22eba8">4. Modern Interpretation of Control Specifications</a>
+<li><a href="#orgba30a55">4. Modern Interpretation of Control Specifications</a>
 <ul>
-<li><a href="#org81a6a17">4.1. Closed Loop Transfer Functions</a></li>
-<li><a href="#org5599054">4.2. Sensitivity Function</a></li>
-<li><a href="#orga4c0eb6">4.3. Robustness: Module Margin</a></li>
-<li><a href="#org69ea990">4.4. Other Requirements</a></li>
+<li><a href="#orgb5fba6b">4.1. Closed Loop Transfer Functions</a></li>
+<li><a href="#orgfccb95f">4.2. Sensitivity Function</a></li>
+<li><a href="#org3a7b849">4.3. Robustness: Module Margin</a></li>
+<li><a href="#orgd23fc15">4.4. Other Requirements</a></li>
 </ul>
 </li>
-<li><a href="#org76a5423">5. \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions</a>
+<li><a href="#org7a2a83d">5. \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions</a>
 <ul>
-<li><a href="#org5eaaa07">5.1. How to Shape closed-loop transfer function? Using Weighting Functions!</a></li>
-<li><a href="#orgea34d7b">5.2. Design of Weighting Functions</a></li>
-<li><a href="#org289ef90">5.3. Shaping the Sensitivity Function</a></li>
-<li><a href="#org8ccbf6c">5.4. Shaping multiple closed-loop transfer functions</a></li>
+<li><a href="#org41e6177">5.1. How to Shape closed-loop transfer function? Using Weighting Functions!</a></li>
+<li><a href="#org48b9767">5.2. Design of Weighting Functions</a></li>
+<li><a href="#org00bea7c">5.3. Shaping the Sensitivity Function</a></li>
+<li><a href="#org5953b0e">5.4. Shaping multiple closed-loop transfer functions</a></li>
 </ul>
 </li>
-<li><a href="#org3865b16">6. Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example</a>
+<li><a href="#org4cfe210">6. Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example</a>
 <ul>
-<li><a href="#org227c600">6.1. Problem</a></li>
-<li><a href="#orge72fa1a">6.2. Typical Procedure</a></li>
-<li><a href="#org2c2c97a">6.3. Step 1 - Shaping of the Sensitivity Function</a></li>
-<li><a href="#org0029581">6.4. Step 2 - Shaping of</a></li>
+<li><a href="#orgbb9bd9a">6.1. Control Problem</a></li>
+<li><a href="#org7f64534">6.2. Control Design Procedure</a></li>
+<li><a href="#orga3ee604">6.3. Step 1 - Shaping of \(S\)</a></li>
+<li><a href="#org238cd32">6.4. Step 2 - Shaping of \(KS\)</a></li>
+<li><a href="#org98f0830">6.5. Step 3 - Shaping of \(T\)</a></li>
 </ul>
 </li>
-<li><a href="#org6864741">7. Conclusion</a></li>
+<li><a href="#orgfd49210">7. Conclusion</a></li>
 </ul>
 </div>
 </div>
 
 <p>
-This document is structured as follows:
+The purpose of this document is to give a <i>practical introduction</i> to the wonderful world of \(\mathcal{H}_\infty\) Control.
+</p>
+
+<p>
+No attend is made to provide an exhaustive treatment of the subject.
+\(\mathcal{H}_\infty\) Control is a very broad topic and entire books are written on it.
+Therefore, for more advanced discussion, please have a look at the recommended references at the bottom of this document.
+</p>
+
+<p>
+When possible, Matlab scripts used for the example/exercises are provided such that you can easily test them on your computer.
+</p>
+
+
+<p>
+The general structure of this document is as follows:
 </p>
 <ul class="org-ul">
-<li>As \(\mathcal{H}_\infty\) Control is a <i>model based</i> control technique, a short introduction to model based control is given in Section <a href="#orga83cb94">1</a></li>
-<li>Classical <i>open</i> loop shaping method is presented in Section <a href="#org1ac1320">2</a>.
-It is also shown that \(\mathcal{H}_\infty\) synthesis can be used for <i>open</i> loop shaping.</li>
-<li>\(\mathcal{H}_\infty\)
-Important concepts such as the \(\mathcal{H}_\infty\) norm and the generalized plant are introduced.</li>
-<li>A</li>
-<li>Finally, an complete example of the
-is performed in Section <a href="#org9c4e967">6</a>.</li>
+<li>A short introduction to <i>model based control</i> is given in Section <a href="#org0e42208">1</a></li>
+<li>Classical <i>open</i> loop shaping method is presented in Section <a href="#org2b3b18e">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="#org0f7ce95">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="#orgb088aa1">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="#orge201bfd">5</a></li>
+<li>Finally, complete examples of the use of \(\mathcal{H}_\infty\) Control for practical problems are provided in Section <a href="#orgb0ce710">6</a>.</li>
 </ul>
 
-<div id="outline-container-org86f37ed" class="outline-2">
-<h2 id="org86f37ed"><span class="section-number-2">1</span> Introduction to Model Based Control</h2>
+<div id="outline-container-org267ac69" class="outline-2">
+<h2 id="org267ac69"><span class="section-number-2">1</span> Introduction to Model Based Control</h2>
 <div class="outline-text-2" id="text-1">
 <p>
-<a id="orga83cb94"></a>
+<a id="org0e42208"></a>
 </p>
 <ul class="org-ul">
-<li>Section <a href="#org8e32f21">1.1</a></li>
-<li>Section <a href="#org77a3c96">1.2</a></li>
-<li>Section <a href="#org88432fd">1.3</a></li>
+<li>Section <a href="#orgf2535d2">1.1</a></li>
+<li>Section <a href="#org3add31e">1.2</a></li>
+<li>Section <a href="#orga75776c">1.3</a></li>
 </ul>
 </div>
 
-<div id="outline-container-org3c879ae" class="outline-3">
-<h3 id="org3c879ae"><span class="section-number-3">1.1</span> Model Based Control - Methodology</h3>
+<div id="outline-container-org2816731" class="outline-3">
+<h3 id="org2816731"><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="org8e32f21"></a>
+<a id="orgf2535d2"></a>
 </p>
 
 <p>
-The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org0d2b0e7">1</a>.
+The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org9cc8abf">1</a>.
 It consists of three steps:
 </p>
 <ol class="org-ol">
@@ -132,7 +148,7 @@ It consists of three steps:
 </ol>
 
 
-<div id="org0d2b0e7" class="figure">
+<div id="org9cc8abf" 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>
@@ -143,24 +159,24 @@ In this document, we will mainly focus on steps 2 and 3.
 </p>
 
 <p>
-Step 2 will be discussed in Section <a href="#orgb49c33d">4</a>.
+Step 2 will be discussed in Section <a href="#orgb088aa1">4</a>.
 There are two main methods for the controller synthesis (step 3):
 </p>
 <ul class="org-ul">
-<li><i>open</i> loop shaping discussed in Section <a href="#org1ac1320">2</a></li>
-<li><i>closed</i> loop shaping discussed in Sections <a href="#orgb49c33d">4</a> and <a href="#org9c4e967">6</a></li>
+<li><i>open</i> loop shaping discussed in Section <a href="#org2b3b18e">2</a></li>
+<li><i>closed</i> loop shaping discussed in Sections <a href="#orgb088aa1">4</a> and <a href="#orgb0ce710">6</a></li>
 </ul>
 </div>
 </div>
 
-<div id="outline-container-org3930ba3" class="outline-3">
-<h3 id="org3930ba3"><span class="section-number-3">1.2</span> From Classical Control to Robust Control</h3>
+<div id="outline-container-org18d6909" class="outline-3">
+<h3 id="org18d6909"><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="org77a3c96"></a>
+<a id="org3add31e"></a>
 </p>
 
-<table id="org51668ba" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org621856b" 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>
@@ -293,7 +309,7 @@ There are two main methods for the controller synthesis (step 3):
 </table>
 
 
-<div id="org822384e" class="figure">
+<div id="orgeef967b" class="figure">
 <p><img src="figs/robustness_performance.png" alt="robustness_performance.png" />
 </p>
 <p><span class="figure-number">Figure 2: </span>Comparison of the performance and robustness of classical control methods, modern control methods and robust control methods. The required information on the plant to succesfuly apply each of the control methods are indicated by the colors.</p>
@@ -301,27 +317,27 @@ There are two main methods for the controller synthesis (step 3):
 </div>
 </div>
 
-<div id="outline-container-orgc80bc10" class="outline-3">
-<h3 id="orgc80bc10"><span class="section-number-3">1.3</span> Example System</h3>
+<div id="outline-container-org4fde82b" class="outline-3">
+<h3 id="org4fde82b"><span class="section-number-3">1.3</span> Example System</h3>
 <div class="outline-text-3" id="text-1-3">
 <p>
-<a id="org88432fd"></a>
+<a id="orga75776c"></a>
 </p>
 
 <p>
-Let&rsquo;s consider the model shown in Figure <a href="#orgeb7716e">3</a>.
+Let&rsquo;s consider the model shown in Figure <a href="#orgd2f0d5a">3</a>.
 It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor.
-The notations used are listed in Table <a href="#orgbf17590">2</a>.
+The notations used are listed in Table <a href="#orgbe269b4">2</a>.
 </p>
 
 
-<div id="orgeb7716e" class="figure">
+<div id="orgd2f0d5a" 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 3: </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="orgbf17590" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="orgbe269b4" 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>
@@ -407,7 +423,7 @@ The notations used are listed in Table <a href="#orgbf17590">2</a>.
 </tbody>
 </table>
 
-<div class="exercice" id="org0d79acb">
+<div class="exercice" id="orgd5c0000">
 <p>
 Derive the following open-loop transfer functions:
 </p>
@@ -438,37 +454,13 @@ You can follow this generic procedure:
 </details>
 
 </div>
-<p>
-Hi Musa,
-Thank you very much for sharing this awesome package.
-For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by &lt;details&gt; blocks.
-</p>
 
 <p>
-For now, I am manually adding #+HTML: &lt;details&gt;&lt;summary&gt;Code&lt;/summary&gt; and #+HTML: &lt;/details&gt; around the source blocks I want to hide&#x2026;
-This is a very simple solution, but not so elegent nor practical.
-</p>
-
-<p>
-Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?
+Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure <a href="#orgd2f0d5a">3</a> into a classical feedback form as shown in Figure <a href="#org7fab7e3">7</a>.
 </p>
 
 
-<p>
-Similarly, I would love to be able to export a &lt;span&gt; block with the name of the file corresponding to the source block.
-For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting!
-</p>
-
-<p>
-Thanks in advance
-</p>
-
-<p>
-Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure <a href="#orgeb7716e">3</a> into a classical feedback form as shown in Figure <a href="#org725d59a">7</a>.
-</p>
-
-
-<div id="org3a9e561" class="figure">
+<div id="org30d0cfb" class="figure">
 <p><img src="figs/classical_feedback_test_system.png" alt="classical_feedback_test_system.png" />
 </p>
 <p><span class="figure-number">Figure 4: </span>Block diagram corresponding to the example system</p>
@@ -486,7 +478,7 @@ Let&rsquo;s define the system parameters on Matlab.
 </div>
 
 <p>
-And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown in Figures <a href="#orgda499f5">5</a> and <a href="#org43d1c16">6</a>).
+And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown in Figures <a href="#org6d94662">5</a> and <a href="#org6752600">6</a>).
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab"><span class="linenr">4: </span>G = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c<span class="org-type">*</span>s <span class="org-type">+</span> k); <span class="org-comment">% Plant</span>
@@ -495,14 +487,14 @@ And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown
 </div>
 
 
-<div id="orgda499f5" class="figure">
+<div id="org6d94662" class="figure">
 <p><img src="figs/bode_plot_example_afm.png" alt="bode_plot_example_afm.png" />
 </p>
 <p><span class="figure-number">Figure 5: </span>Bode plot of the plant \(G(s)\)</p>
 </div>
 
 
-<div id="org43d1c16" class="figure">
+<div id="org6752600" class="figure">
 <p><img src="figs/bode_plot_example_Gd.png" alt="bode_plot_example_Gd.png" />
 </p>
 <p><span class="figure-number">Figure 6: </span>Magnitude of the disturbance transfer function \(G_d(s)\)</p>
@@ -511,44 +503,44 @@ And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown
 </div>
 </div>
 
-<div id="outline-container-org5e5539a" class="outline-2">
-<h2 id="org5e5539a"><span class="section-number-2">2</span> Classical Open Loop Shaping</h2>
+<div id="outline-container-org1bfaa00" class="outline-2">
+<h2 id="org1bfaa00"><span class="section-number-2">2</span> Classical Open Loop Shaping</h2>
 <div class="outline-text-2" id="text-2">
 <p>
-<a id="org1ac1320"></a>
+<a id="org2b3b18e"></a>
 </p>
 <ul class="org-ul">
-<li>Section <a href="#org7166a19">2.1</a></li>
-<li>Section <a href="#org2157a8f">2.2</a></li>
-<li>Section <a href="#org6f1355b">2.3</a></li>
-<li>Section <a href="#orga101e0c">2.4</a></li>
+<li>Section <a href="#org2ee971f">2.1</a></li>
+<li>Section <a href="#orgf72f718">2.2</a></li>
+<li>Section <a href="#org0f2dc6c">2.3</a></li>
+<li>Section <a href="#orgb02a40a">2.4</a></li>
 </ul>
 </div>
 
-<div id="outline-container-orgacb6d03" class="outline-3">
-<h3 id="orgacb6d03"><span class="section-number-3">2.1</span> Introduction to Loop Shaping</h3>
+<div id="outline-container-org4d52f02" class="outline-3">
+<h3 id="org4d52f02"><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="org7166a19"></a>
+<a id="org2ee971f"></a>
 </p>
 
-<div class="definition" id="orgd0b3e20">
+<div class="definition" id="orgd605846">
 <p>
 <b>Loop Shaping</b> refers to a design procedure that involves explicitly shaping the magnitude of the <b>Loop Transfer Function</b> \(L(s)\).
 </p>
 
 </div>
 
-<div class="definition" id="org4f34bd2">
+<div class="definition" id="orgba76f81">
 <p>
-The <b>Loop Gain</b> \(L(s)\) usually refers to as the product of the controller and the plant (&ldquo;Gain around the loop&rdquo;, see Figure <a href="#org725d59a">7</a>):
+The <b>Loop Gain</b> \(L(s)\) usually refers to as the product of the controller and the plant (&ldquo;Gain around the loop&rdquo;, see Figure <a href="#org7fab7e3">7</a>):
 </p>
 \begin{equation}
   L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
 \end{equation}
 
 
-<div id="org725d59a" class="figure">
+<div id="org7fab7e3" class="figure">
 <p><img src="figs/open_loop_shaping.png" alt="open_loop_shaping.png" />
 </p>
 <p><span class="figure-number">Figure 7: </span>Classical Feedback Architecture</p>
@@ -573,11 +565,11 @@ The Open Loop shape is usually done manually has the loop gain \(L(s)\) depends
 </p>
 
 <p>
-\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that \(L(s)\) has the wanted shape (an example is shown in Figure <a href="#orgf9755ec">8</a>).
+\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that \(L(s)\) has the wanted shape (an example is shown in Figure <a href="#org6318e13">8</a>).
 </p>
 
 
-<div id="orgf9755ec" class="figure">
+<div id="org6318e13" class="figure">
 <p><img src="figs/open_loop_shaping_shape.png" alt="open_loop_shaping_shape.png" />
 </p>
 <p><span class="figure-number">Figure 8: </span>Typical Wanted Shape for the Loop Gain \(L(s)\)</p>
@@ -585,14 +577,14 @@ The Open Loop shape is usually done manually has the loop gain \(L(s)\) depends
 </div>
 </div>
 
-<div id="outline-container-orgfac68ae" class="outline-3">
-<h3 id="orgfac68ae"><span class="section-number-3">2.2</span> Example of Open Loop Shaping</h3>
+<div id="outline-container-orgfefb3c0" class="outline-3">
+<h3 id="orgfefb3c0"><span class="section-number-3">2.2</span> Example of Open Loop Shaping</h3>
 <div class="outline-text-3" id="text-2-2">
 <p>
-<a id="org2157a8f"></a>
+<a id="orgf72f718"></a>
 </p>
 
-<div class="exampl" id="org04b7fa2">
+<div class="exampl" id="org393716f">
 <p>
 Let&rsquo;s take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
 </p>
@@ -604,7 +596,7 @@ Let&rsquo;s take our example system and try to apply the Open-Loop shaping strat
 
 </div>
 
-<div class="exercice" id="orgbf8019f">
+<div class="exercice" id="orgebd2d6a">
 <p>
 Using <code>SISOTOOL</code>, design a controller that fulfill the specifications.
 </p>
@@ -621,7 +613,7 @@ In order to have the wanted Roll-off, two integrators are used, a lead is also a
 </p>
 
 <p>
-The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure <a href="#org7a79600">9</a>.
+The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure <a href="#org6c9a891">9</a>.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">K = 14e8 <span class="org-type">*</span> ...<span class="org-comment"> % Gain</span>
@@ -632,7 +624,7 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
 </div>
 
 
-<div id="org7a79600" class="figure">
+<div id="org6c9a891" class="figure">
 <p><img src="figs/loop_gain_manual_afm.png" alt="loop_gain_manual_afm.png" />
 </p>
 <p><span class="figure-number">Figure 9: </span>Bode Plot of the obtained Loop Gain \(L(s) = G(s) K(s)\)</p>
@@ -680,11 +672,11 @@ And we can verify that we have the wanted stability margins:
 </div>
 </div>
 
-<div id="outline-container-orgcc18a4a" class="outline-3">
-<h3 id="orgcc18a4a"><span class="section-number-3">2.3</span> \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
+<div id="outline-container-org6fdc1bc" class="outline-3">
+<h3 id="org6fdc1bc"><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="org6f1355b"></a>
+<a id="org0f2dc6c"></a>
 </p>
 
 <p>
@@ -711,7 +703,7 @@ where:
 <li><code>K</code> is the synthesize controller</li>
 </ul>
 
-<div class="seealso" id="org06a256d">
+<div class="seealso" id="org7913b44">
 <p>
 Matlab documentation of <code>loopsyn</code> (<a href="https://www.mathworks.com/help/robust/ref/loopsyn.html">link</a>).
 </p>
@@ -720,11 +712,11 @@ Matlab documentation of <code>loopsyn</code> (<a href="https://www.mathworks.com
 </div>
 </div>
 
-<div id="outline-container-orgc6b8e73" class="outline-3">
-<h3 id="orgc6b8e73"><span class="section-number-3">2.4</span> Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
+<div id="outline-container-org1172944" class="outline-3">
+<h3 id="org1172944"><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="orga101e0c"></a>
+<a id="orgb02a40a"></a>
 </p>
 
 <p>
@@ -755,7 +747,7 @@ The \(\mathcal{H}_\infty\) optimal open loop shaping synthesis is performed usin
 </pre>
 </div>
 
-<div class="important" id="org144dc06">
+<div class="important" id="orgdea5110">
 <p>
 It is always important to analyze the controller after the synthesis is performed.
 </p>
@@ -767,7 +759,7 @@ In the end, a synthesize controller is just a combination of low pass filters, h
 </div>
 
 <p>
-Let&rsquo;s briefly analyze the obtained controller which bode plot is shown in Figure <a href="#org0b6eab6">10</a>:
+Let&rsquo;s briefly analyze the obtained controller which bode plot is shown in Figure <a href="#orgd37b625">10</a>:
 </p>
 <ul class="org-ul">
 <li>two integrators are used at low frequency to have the wanted low frequency high gain</li>
@@ -776,28 +768,28 @@ Let&rsquo;s briefly analyze the obtained controller which bode plot is shown in
 </ul>
 
 
-<div id="org0b6eab6" class="figure">
+<div id="orgd37b625" 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>
 </div>
 
 <p>
-The obtained Loop Gain is shown in Figure <a href="#org64b8397">11</a> and matches the specified one by a factor \(\gamma\).
+The obtained Loop Gain is shown in Figure <a href="#orgc539656">11</a> and matches the specified one by a factor \(\gamma\).
 </p>
 
 
-<div id="org64b8397" class="figure">
+<div id="orgc539656" 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 11: </span>Obtained Open Loop Gain \(L(s) = G(s) K(s)\) and comparison with the wanted Loop gain \(L_w\)</p>
 </div>
 
 <p>
-Let&rsquo;s now compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table <a href="#org061cbfe">3</a>.
+Let&rsquo;s now compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table <a href="#org125376e">3</a>.
 </p>
 
-<table id="org061cbfe" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org125376e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 3:</span> Comparison of the characteristics obtained with the two methods</caption>
 
 <colgroup>
@@ -838,29 +830,29 @@ Let&rsquo;s now compare the obtained stability margins of the \(\mathcal{H}_\inf
 </div>
 </div>
 
-<div id="outline-container-org397a600" class="outline-2">
-<h2 id="org397a600"><span class="section-number-2">3</span> A first Step into the \(\mathcal{H}_\infty\) world</h2>
+<div id="outline-container-orgde10b7a" class="outline-2">
+<h2 id="orgde10b7a"><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="org463c583"></a>
+<a id="org0f7ce95"></a>
 </p>
 <ul class="org-ul">
-<li>Section <a href="#org2523d10">3.1</a></li>
-<li>Section <a href="#orgc15fa44">3.2</a></li>
-<li>Section <a href="#org953705e">3.3</a></li>
-<li>Section <a href="#org0b20855">3.4</a></li>
-<li>Section <a href="#org7bc9c12">3.5</a></li>
+<li>Section <a href="#orgac50119">3.1</a></li>
+<li>Section <a href="#orgcee36df">3.2</a></li>
+<li>Section <a href="#org0dd62f4">3.3</a></li>
+<li>Section <a href="#org58e1225">3.4</a></li>
+<li>Section <a href="#orga9839ce">3.5</a></li>
 </ul>
 </div>
 
-<div id="outline-container-org60d66b0" class="outline-3">
-<h3 id="org60d66b0"><span class="section-number-3">3.1</span> The \(\mathcal{H}_\infty\) Norm</h3>
+<div id="outline-container-org5ae8918" class="outline-3">
+<h3 id="org5ae8918"><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="org2523d10"></a>
+<a id="orgac50119"></a>
 </p>
 
-<div class="definition" id="orga58707a">
+<div class="definition" id="org05b3799">
 <p>
 The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
 </p>
@@ -877,7 +869,7 @@ For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a
 
 </div>
 
-<div class="exampl" id="org8925151">
+<div class="exampl" id="orgf6e57d9">
 <p>
 Let&rsquo;s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) using the <code>hinfnorm</code> function:
 </p>
@@ -892,11 +884,11 @@ Let&rsquo;s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) u
 
 
 <p>
-We can see that the \(\mathcal{H}_\infty\) norm of \(G(s)\) does corresponds to the peak value of \(|G(j\omega)|\) as a function of frequency as shown in Figure <a href="#org85cb7a1">12</a>.
+We can see that the \(\mathcal{H}_\infty\) norm of \(G(s)\) does corresponds to the peak value of \(|G(j\omega)|\) as a function of frequency as shown in Figure <a href="#orgb535058">12</a>.
 </p>
 
 
-<div id="org85cb7a1" class="figure">
+<div id="orgb535058" class="figure">
 <p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" />
 </p>
 <p><span class="figure-number">Figure 12: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p>
@@ -906,14 +898,14 @@ We can see that the \(\mathcal{H}_\infty\) norm of \(G(s)\) does corresponds to
 </div>
 </div>
 
-<div id="outline-container-org66d736a" class="outline-3">
-<h3 id="org66d736a"><span class="section-number-3">3.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
+<div id="outline-container-org91b1ae9" class="outline-3">
+<h3 id="org91b1ae9"><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="orgc15fa44"></a>
+<a id="orgcee36df"></a>
 </p>
 
-<div class="definition" id="orga689541">
+<div class="definition" id="org5b4c588">
 <p>
 \(\mathcal{H}_\infty\) synthesis is a method that uses an <b>algorithm</b> (LMI optimization, Riccati equation) to find a controller that stabilize the system and that <b>minimizes</b> the \(\mathcal{H}_\infty\) norms of defined transfer functions.
 </p>
@@ -949,11 +941,11 @@ Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
 </div>
 </div>
 
-<div id="outline-container-org7054eb2" class="outline-3">
-<h3 id="org7054eb2"><span class="section-number-3">3.3</span> The Generalized Plant</h3>
+<div id="outline-container-org9c4b033" class="outline-3">
+<h3 id="org9c4b033"><span class="section-number-3">3.3</span> The Generalized Plant</h3>
 <div class="outline-text-3" id="text-3-3">
 <p>
-<a id="org953705e"></a>
+<a id="org0dd62f4"></a>
 </p>
 
 <p>
@@ -963,9 +955,9 @@ It makes things much easier for the following steps.
 </p>
 
 <p>
-The generalized plant, usually noted \(P(s)\), is shown in Figure <a href="#orgfd30978">13</a>.
+The generalized plant, usually noted \(P(s)\), is shown in Figure <a href="#orga8dfa5c">13</a>.
 It has two inputs and two outputs (both could contains many signals).
-The meaning of the inputs and outputs are summarized in Table <a href="#org97dc6e9">4</a>.
+The meaning of the inputs and outputs are summarized in Table <a href="#org5801c5b">4</a>.
 </p>
 
 <p>
@@ -978,14 +970,14 @@ It can indeed represent feedback as well as feedforward control architectures.
 \end{equation}
 
 
-<div id="orgfd30978" class="figure">
+<div id="orga8dfa5c" class="figure">
 <p><img src="figs/general_plant.png" alt="general_plant.png" />
 </p>
 <p><span class="figure-number">Figure 13: </span>Inputs and Outputs of the generalized Plant</p>
 </div>
 
-<div class="important" id="org9943179">
-<table id="org97dc6e9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<div class="important" id="org01ed6f0">
+<table id="org5801c5b" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 4:</span> Notations for the general configuration</caption>
 
 <colgroup>
@@ -1031,18 +1023,18 @@ It can indeed represent feedback as well as feedforward control architectures.
 </div>
 </div>
 
-<div id="outline-container-orga725c9e" class="outline-3">
-<h3 id="orga725c9e"><span class="section-number-3">3.4</span> The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant</h3>
+<div id="outline-container-org0187c7b" class="outline-3">
+<h3 id="org0187c7b"><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="org0b20855"></a>
+<a id="org58e1225"></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="org23d6a26">
+<div class="important" id="org67a0e05">
 <dl class="org-dl">
 <dt>\(\mathcal{H}_\infty\) Synthesis applied on the generalized plant</dt><dd></dd>
 </dl>
@@ -1057,7 +1049,7 @@ After \(K\) is found, the system is <i>robustified</i> by adjusting the response
 </div>
 
 
-<div id="org7a14eee" class="figure">
+<div id="orgc77ed12" class="figure">
 <p><img src="figs/general_control_names.png" alt="general_control_names.png" />
 </p>
 <p><span class="figure-number">Figure 14: </span>General Control Configuration</p>
@@ -1089,28 +1081,29 @@ where:
 </div>
 </div>
 
-<div id="outline-container-orgfec6fc7" class="outline-3">
-<h3 id="orgfec6fc7"><span class="section-number-3">3.5</span> From a Classical Feedback Architecture to a Generalized Plant</h3>
+<div id="outline-container-org822d770" class="outline-3">
+<h3 id="org822d770"><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="org7bc9c12"></a>
+<a id="orga9839ce"></a>
 </p>
 
 <p>
-The procedure to convert a typical control architecture as the one shown in Figure <a href="#orgfd2df1c">15</a> to a generalized Plant is as follows:
+The procedure to convert a typical control architecture as the one shown in Figure <a href="#orgb770b94">15</a> to a generalized Plant is as follows:
 </p>
 <ol class="org-ol">
 <li>Define signals (\(w\), \(z\), \(u\) and \(v\)) of the generalized plant</li>
-<li>Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration</li>
+<li>Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration shown in Figure <a href="#orga8dfa5c">13</a></li>
 </ol>
 
-<div class="exercice" id="org9b29734">
-<p>
-Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgfd2df1c">15</a>
-</p>
+<div class="exercice" id="org7736562">
+<ol class="org-ol">
+<li>Convert the tracking control architecture shown in Figure <a href="#orgb770b94">15</a> to a generalized configuration</li>
+<li>Compute the transfer function matrix using Matlab as a function or \(K\) and \(G\)</li>
+</ol>
 
 
-<div id="orgfd2df1c" class="figure">
+<div id="orgb770b94" class="figure">
 <p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" />
 </p>
 <p><span class="figure-number">Figure 15: </span>Classical Feedback Control Architecture (Tracking)</p>
@@ -1122,29 +1115,27 @@ First, define the signals of the generalized plant:
 </p>
 <ul class="org-ul">
 <li>Exogenous inputs: \(w = r\)</li>
-<li>Signals to be minimized: \(z_1 = \epsilon\), \(z_2 = u\)</li>
-<li>Control signals: \(v = y\)</li>
-<li>Control inputs: \(u\)</li>
+<li>Signals to be minimized:
+Usually, we want to minimize the tracking errors \(\epsilon\) and the control signal \(u\): \(z = [\epsilon,\ u]\)</li>
+<li>Controller inputs: this is the signal at the input of the controller: \(v = \epsilon\)</li>
+<li>Control inputs: signal generated by the controller: \(u\)</li>
 </ul>
 
 <p>
-Then, Remove \(K\) and rearrange the inputs and outputs.
+Then, Remove \(K\) and rearrange the inputs and outputs as in Figure <a href="#orga8dfa5c">13</a>.
 </p>
 </details>
 
 <details><summary>Answer</summary>
 <p>
-The obtained generalized plant shown in Figure <a href="#org11763c3">16</a>.
+The obtained generalized plant shown in Figure <a href="#orgeae5848">16</a>.
 </p>
 
 
-<div id="org11763c3" class="figure">
+<div id="orgeae5848" class="figure">
 <p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" />
 </p>
 <p><span class="figure-number">Figure 16: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p>
-</div>
-</details>
-
 </div>
 
 <p>
@@ -1157,26 +1148,29 @@ Using Matlab, the generalized plant can be defined as follows:
 P.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'u'</span>};
 P.OutputName = {<span class="org-string">'e'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>};
 </pre>
+</div>
+</details>
+
 </div>
 </div>
 </div>
 </div>
 
-<div id="outline-container-orgb22eba8" class="outline-2">
-<h2 id="orgb22eba8"><span class="section-number-2">4</span> Modern Interpretation of Control Specifications</h2>
+<div id="outline-container-orgba30a55" class="outline-2">
+<h2 id="orgba30a55"><span class="section-number-2">4</span> Modern Interpretation of Control Specifications</h2>
 <div class="outline-text-2" id="text-4">
 <p>
-<a id="orgb49c33d"></a>
+<a id="orgb088aa1"></a>
 </p>
 <ul class="org-ul">
-<li>Section <a href="#org72c7250">4.1</a></li>
-<li>Section <a href="#orga4cc13f">4.2</a></li>
-<li>Section <a href="#org1c3d094">4.3</a></li>
-<li>Section <a href="#orgc2e1843">4.4</a></li>
+<li>Section <a href="#org62393f7">4.1</a></li>
+<li>Section <a href="#org031eafd">4.2</a></li>
+<li>Section <a href="#org48eee4f">4.3</a></li>
+<li>Section <a href="#org58a8ffb">4.4</a></li>
 </ul>
 
 <p>
-As shown in Section <a href="#org1ac1320">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="#org2b3b18e">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>
@@ -1186,11 +1180,11 @@ 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="#orgedfe642">17</a>, we can link to the following specifications to closed-loop transfer functions.
-This is summarized in Table <a href="#orgcf21b30">5</a>.
+If we consider the feedback system shown in Figure <a href="#org3b8b88c">17</a>, we can link to the following specifications to closed-loop transfer functions.
+This is summarized in Table <a href="#org0b03397">5</a>.
 </p>
 
-<table id="orgcf21b30" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org0b03397" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 5:</span> Typical Specification and associated closed-loop transfer function</caption>
 
 <colgroup>
@@ -1232,33 +1226,33 @@ This is summarized in Table <a href="#orgcf21b30">5</a>.
 
 <tr>
 <td class="org-left">Robustness (stability margins)</td>
-<td class="org-left">Module margin (see Section <a href="#org1c3d094">4.3</a>)</td>
+<td class="org-left">Module margin (see Section <a href="#org48eee4f">4.3</a>)</td>
 </tr>
 </tbody>
 </table>
 
 
-<div id="orgedfe642" class="figure">
+<div id="org3b8b88c" class="figure">
 <p><img src="figs/gang_of_four_feedback.png" alt="gang_of_four_feedback.png" />
 </p>
 <p><span class="figure-number">Figure 17: </span>Simple Feedback Architecture</p>
 </div>
 </div>
 
-<div id="outline-container-org81a6a17" class="outline-3">
-<h3 id="org81a6a17"><span class="section-number-3">4.1</span> Closed Loop Transfer Functions</h3>
+<div id="outline-container-orgb5fba6b" class="outline-3">
+<h3 id="orgb5fba6b"><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="org72c7250"></a>
+<a id="org62393f7"></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="org0eb900d">
+<div class="exercice" id="org487a8a6">
 <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="#orgedfe642">17</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="#org3b8b88c">17</a>.
 </p>
 
 <details><summary>Hint</summary>
@@ -1295,9 +1289,9 @@ The following equations should be obtained:
 
 </div>
 
-<div class="important" id="org015d851">
+<div class="important" id="org608c9e5">
 <p>
-We can see that they are 4 different transfer functions describing the behavior of the system in Figure <a href="#orgedfe642">17</a>.
+We can see that they are 4 different transfer functions describing the behavior of the system in Figure <a href="#org3b8b88c">17</a>.
 These called the <b>Gang of Four</b>:
 </p>
 \begin{align}
@@ -1309,7 +1303,7 @@ These called the <b>Gang of Four</b>:
 
 </div>
 
-<div class="seealso" id="org4167b45">
+<div class="seealso" id="orgb7ed2ba">
 <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>.
@@ -1333,11 +1327,11 @@ Similarly, to reduce the effect of measurement noise \(n\) on the output \(y\),
 </div>
 </div>
 
-<div id="outline-container-org5599054" class="outline-3">
-<h3 id="org5599054"><span class="section-number-3">4.2</span> Sensitivity Function</h3>
+<div id="outline-container-orgfccb95f" class="outline-3">
+<h3 id="orgfccb95f"><span class="section-number-3">4.2</span> Sensitivity Function</h3>
 <div class="outline-text-3" id="text-4-2">
 <p>
-<a id="orga4cc13f"></a>
+<a id="org031eafd"></a>
 </p>
 
 <p>
@@ -1348,14 +1342,14 @@ 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="#org60b4fea">18</a> and the corresponding step responses are shown in Figure <a href="#org8b40a7d">19</a>.
+The obtained sensitivity functions are shown in Figure <a href="#orge015436">18</a> and the corresponding step responses are shown in Figure <a href="#orgf89947c">19</a>.
 </p>
 
 <p>
-The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table <a href="#org01c0c70">6</a>.
+The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table <a href="#org4e8e9ce">6</a>.
 </p>
 
-<table id="org01c0c70" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org4e8e9ce" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 6:</span> Comparison of the sensitivity function shape and the corresponding step response for the three controller variations</caption>
 
 <colgroup>
@@ -1394,20 +1388,20 @@ The comparison of the sensitivity functions shapes and their effect on the step
 </table>
 
 
-<div id="org60b4fea" class="figure">
+<div id="orge015436" class="figure">
 <p><img src="figs/sensitivity_shape_effect.png" alt="sensitivity_shape_effect.png" />
 </p>
 <p><span class="figure-number">Figure 18: </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="org8b40a7d" class="figure">
+<div id="orgf89947c" class="figure">
 <p><img src="figs/sensitivity_shape_effect_step.png" alt="sensitivity_shape_effect_step.png" />
 </p>
 <p><span class="figure-number">Figure 19: </span>Step response (response from \(r\) to \(y\)) for the different controllers</p>
 </div>
 
-<div class="definition" id="orga231766">
+<div class="definition" id="org1ac5204">
 <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.
@@ -1420,9 +1414,9 @@ In general, a large bandwidth corresponds to a faster rise time.
 
 </div>
 
-<div class="important" id="orgc56bfc1">
+<div class="important" id="org3d8422a">
 <p>
-From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure <a href="#org5478b9b">20</a>.
+From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure <a href="#org8b71a00">20</a>.
 </p>
 
 <p>
@@ -1437,7 +1431,7 @@ This will become clear in the next section about the <b>module margin</b>.</li>
 </ul>
 
 
-<div id="org5478b9b" class="figure">
+<div id="org8b71a00" class="figure">
 <p><img src="figs/h-infinity-spec-S.png" alt="h-infinity-spec-S.png" />
 </p>
 <p><span class="figure-number">Figure 20: </span>Typical wanted shape of the Sensitivity transfer function</p>
@@ -1447,18 +1441,18 @@ This will become clear in the next section about the <b>module margin</b>.</li>
 </div>
 </div>
 
-<div id="outline-container-orga4c0eb6" class="outline-3">
-<h3 id="orga4c0eb6"><span class="section-number-3">4.3</span> Robustness: Module Margin</h3>
+<div id="outline-container-org3a7b849" class="outline-3">
+<h3 id="org3a7b849"><span class="section-number-3">4.3</span> Robustness: Module Margin</h3>
 <div class="outline-text-3" id="text-4-3">
 <p>
-<a id="org1c3d094"></a>
+<a id="org48eee4f"></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="org90c78f4">
+<div class="exampl" id="org45b1a22">
 <p>
 Let&rsquo;s consider the following plant \(G_t(s)\):
 </p>
@@ -1472,7 +1466,7 @@ 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="#org5acd1bc">21</a>.
+Let&rsquo;s say we have designed a controller \(K_t(s)\) that gives the loop gain shown in Figure <a href="#org1442363">21</a>.
 </p>
 
 <div class="org-src-container">
@@ -1481,7 +1475,7 @@ 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="#org5acd1bc">21</a>:
+The following characteristics can be determined from Figure <a href="#org1442363">21</a>:
 </p>
 <ul class="org-ul">
 <li>bandwidth of \(\approx 10\, \text{Hz}\)</li>
@@ -1494,7 +1488,7 @@ This might indicate very good robustness properties of the closed-loop system.
 </p>
 
 
-<div id="org5acd1bc" class="figure">
+<div id="org1442363" 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 21: </span>Bode plot of the obtained Loop Gain \(L(s)\)</p>
@@ -1510,7 +1504,7 @@ 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="#org7224cfc">22</a>.
+The obtained &ldquo;real&rdquo; loop gain is shown in Figure <a href="#org85d0b3c">22</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>
 
@@ -1528,7 +1522,7 @@ It is confirmed by checking the stability of the closed loop system:
 
 
 
-<div id="org7224cfc" class="figure">
+<div id="org85d0b3c" 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 22: </span>Bode plots of \(L(s)\) (loop gain corresponding the nominal plant) and \(L_r(s)\) (loop gain corresponding to the real plant)</p>
@@ -1548,7 +1542,7 @@ 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="org42ccdcc">
+<div class="definition" id="org4f95cb7">
 <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>
 
@@ -1557,7 +1551,7 @@ Let&rsquo;s now determine a new robustness indicator based on the Nyquist Stabil
 
 </div>
 
-<div class="seealso" id="org4d08192">
+<div class="seealso" id="org3426c8f">
 <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>
@@ -1569,21 +1563,21 @@ 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="orgead502a">
+<div class="definition" id="orga17372e">
 <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="org34eb152">
+<div class="exampl" id="org66221fa">
 <p>
-A typical Nyquist plot is shown in Figure <a href="#orge124ea5">23</a>.
+A typical Nyquist plot is shown in Figure <a href="#org11a616e">23</a>.
 The gain, phase and module margins are graphically shown to have an idea of what they represent.
 </p>
 
 
-<div id="orge124ea5" class="figure">
+<div id="org11a616e" class="figure">
 <p><img src="figs/module_margin_example.png" alt="module_margin_example.png" />
 </p>
 <p><span class="figure-number">Figure 23: </span>Nyquist plot with visual indication of the Gain margin \(\Delta G\), Phase margin \(\Delta \phi\) and Module margin \(\Delta M\)</p>
@@ -1592,7 +1586,7 @@ The gain, phase and module margins are graphically shown to have an idea of what
 </div>
 
 <p>
-As expected from Figure <a href="#orge124ea5">23</a>, there is a close relationship between the module margin and the gain and phase margins.
+As expected from Figure <a href="#org11a616e">23</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}
@@ -1611,7 +1605,7 @@ 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="org1f3f902">
+<div class="important" id="org84495d5">
 <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>
@@ -1631,14 +1625,14 @@ 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="org0b144cc">
+<div class="exampl" id="org0a89e20">
 <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="orgc7d1d4a">
+<div class="seealso" id="org416271e">
 <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>
@@ -1647,11 +1641,11 @@ To learn more about module/disk margin, you can check out <a href="https://www.y
 </div>
 </div>
 
-<div id="outline-container-org69ea990" class="outline-3">
-<h3 id="org69ea990"><span class="section-number-3">4.4</span> Other Requirements</h3>
+<div id="outline-container-orgd23fc15" class="outline-3">
+<h3 id="orgd23fc15"><span class="section-number-3">4.4</span> Other Requirements</h3>
 <div class="outline-text-3" id="text-4-4">
 <p>
-<a id="orgc2e1843"></a>
+<a id="org58a8ffb"></a>
 </p>
 
 <p>
@@ -1680,7 +1674,7 @@ with:
 Noise Attenuation: typical wanted shape for \(T\)
 </p>
 
-<table id="orgecd685d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org298b4d4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 7:</span> Typical Specifications and corresponding wanted norms of open and closed loop tansfer functions</caption>
 
 <colgroup>
@@ -1733,11 +1727,11 @@ Noise Attenuation: typical wanted shape for \(T\)
 </div>
 </div>
 
-<div id="outline-container-org76a5423" class="outline-2">
-<h2 id="org76a5423"><span class="section-number-2">5</span> \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions</h2>
+<div id="outline-container-org7a2a83d" class="outline-2">
+<h2 id="org7a2a83d"><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="orgd958131"></a>
+<a id="orge201bfd"></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.
@@ -1754,22 +1748,22 @@ But don&rsquo;t worry, the \(\mathcal{H}_\infty\) synthesis will do this job for
 
 <p>
 This
-Section <a href="#org5ed9a9d">5.1</a>
-Section <a href="#org858ec33">5.2</a>
-Section <a href="#orgd2cf49d">5.3</a>
-Section <a href="#org7ad7325">5.4</a>
+Section <a href="#org01843d4">5.1</a>
+Section <a href="#orgb646bda">5.2</a>
+Section <a href="#org0a34d87">5.3</a>
+Section <a href="#org32ad878">5.4</a>
 </p>
 </div>
 
-<div id="outline-container-org5eaaa07" class="outline-3">
-<h3 id="org5eaaa07"><span class="section-number-3">5.1</span> How to Shape closed-loop transfer function? Using Weighting Functions!</h3>
+<div id="outline-container-org41e6177" class="outline-3">
+<h3 id="org41e6177"><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="org5ed9a9d"></a>
+<a id="org01843d4"></a>
 </p>
 
 <p>
-If the \(\mathcal{H}_\infty\) synthesis is applied on the generalized plant \(P(s)\) shown in Figure <a href="#org0cab7a5">24</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="#org2ce660a">24</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>
@@ -1779,16 +1773,16 @@ However, as the \(\mathcal{H}_\infty\) norm is the maximum peak value of the tra
 </p>
 
 
-<div id="org0cab7a5" class="figure">
+<div id="org2ce660a" 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 24: </span>Generalized Plant</p>
 </div>
 
 
-<div class="important" id="orgcddbc92">
+<div class="important" id="org39128d0">
 <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="#org2fb918f">25</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="#org4b7c49f">25</a>.
 </p>
 
 <p>
@@ -1806,7 +1800,7 @@ 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="org45410de">
+<div class="important" id="orgb1eb251">
 <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>
@@ -1818,15 +1812,15 @@ Therefore, the choice of the weighting function \(W_s(s)\) is very important: it
 </div>
 
 
-<div id="org2fb918f" class="figure">
+<div id="org4b7c49f" 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 25: </span>Weighted Generalized Plant</p>
 </div>
 
-<div class="exercice" id="orgd7d896e">
+<div class="exercice" id="org7d46e53">
 <p>
-Using matlab, compute the weighted generalized plant shown in Figure <a href="#orgeeaf03f">26</a> as a function of \(G(s)\) and \(W_S(s)\).
+Using matlab, compute the weighted generalized plant shown in Figure <a href="#org0dd0c0f">26</a> as a function of \(G(s)\) and \(W_S(s)\).
 </p>
 
 <details><summary>Hint</summary>
@@ -1864,18 +1858,18 @@ The second solution is however more general, and can also be used when weights a
 </div>
 </div>
 
-<div id="outline-container-orgea34d7b" class="outline-3">
-<h3 id="orgea34d7b"><span class="section-number-3">5.2</span> Design of Weighting Functions</h3>
+<div id="outline-container-org48b9767" class="outline-3">
+<h3 id="org48b9767"><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="org858ec33"></a>
+<a id="orgb646bda"></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="orgb23d76c">
+<div class="definition" id="org3518a83">
 <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>
@@ -1900,9 +1894,9 @@ with:
 <li><code>hfgain</code>: high frequency gain</li>
 </ul>
 
-<div class="exampl" id="org2031a72">
+<div class="exampl" id="org7ac321e">
 <p>
-The Matlab code below produces a weighting function with the following characteristics (Figure <a href="#orgeeaf03f">26</a>):
+The Matlab code below produces a weighting function with the following characteristics (Figure <a href="#org0dd0c0f">26</a>):
 </p>
 <ul class="org-ul">
 <li>Low frequency gain of 100</li>
@@ -1916,7 +1910,7 @@ The Matlab code below produces a weighting function with the following character
 </div>
 
 
-<div id="orgeeaf03f" class="figure">
+<div id="org0dd0c0f" class="figure">
 <p><img src="figs/first_order_weight.png" alt="first_order_weight.png" />
 </p>
 <p><span class="figure-number">Figure 26: </span>Obtained Magnitude of the Weighting Function</p>
@@ -1924,7 +1918,7 @@ The Matlab code below produces a weighting function with the following character
 
 </div>
 
-<div class="seealso" id="org068939f">
+<div class="seealso" id="org9e665ab">
 <p>
 Quite often, higher orders weights are required.
 </p>
@@ -1956,7 +1950,7 @@ 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="org3a88912"><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="org286a722"><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
@@ -1991,11 +1985,11 @@ W3 = generateWeight(<span class="org-string">'G0'</span>, 1e2, <span class="org-
 </div>
 
 <p>
-The obtained shapes are shown in Figure <a href="#orgb12139b">27</a>.
+The obtained shapes are shown in Figure <a href="#org0b8a200">27</a>.
 </p>
 
 
-<div id="orgb12139b" class="figure">
+<div id="org0b8a200" class="figure">
 <p><img src="figs/high_order_weight.png" alt="high_order_weight.png" />
 </p>
 <p><span class="figure-number">Figure 27: </span>Higher order weights using Equation \eqref{eq:weight_formula_advanced}</p>
@@ -2005,11 +1999,11 @@ The obtained shapes are shown in Figure <a href="#orgb12139b">27</a>.
 </div>
 </div>
 
-<div id="outline-container-org289ef90" class="outline-3">
-<h3 id="org289ef90"><span class="section-number-3">5.3</span> Shaping the Sensitivity Function</h3>
+<div id="outline-container-org00bea7c" class="outline-3">
+<h3 id="org00bea7c"><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="orgd2cf49d"></a>
+<a id="org0a34d87"></a>
 </p>
 
 <p>
@@ -2022,17 +2016,17 @@ 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="#org88432fd">1.3</a>.
+As usual, the plant used is the one presented in Section <a href="#orga75776c">1.3</a>.
 </p>
 
-<div class="exercice" id="orgfca1502">
+<div class="exercice" id="orga0248cc">
 <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="#org5f95be2">28</a>.
+The typical wanted upper bound of the sensitivity function is shown in Figure <a href="#orgb9e3a64">28</a>.
 </p>
 
 <p>
@@ -2050,7 +2044,7 @@ Remember that the wanted upper bound of the sensitivity function is defined by t
 </p>
 
 
-<div id="org5f95be2" class="figure">
+<div id="orgb9e3a64" class="figure">
 <p><img src="figs/h-infinity-spec-S.png" alt="h-infinity-spec-S.png" />
 </p>
 <p><span class="figure-number">Figure 28: </span>Typical wanted shape of the Sensitivity transfer function</p>
@@ -2115,7 +2109,7 @@ And the \(\mathcal{H}_\infty\) synthesis is performed on the <i>weighted</i> gen
 </pre>
 </div>
 
-<pre class="example" id="org3e8a5d2">
+<pre class="example" id="org81ed00f">
 Test bounds:  0.5 &lt;=  gamma  &lt;=  0.51
 
  gamma        X&gt;=0        Y&gt;=0       rho(XY)&lt;1    p/f
@@ -2136,10 +2130,10 @@ 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="#org6a4f956">29</a>.
+This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure <a href="#org4a1fd1a">29</a>.
 </p>
 
-<div class="important" id="org95d83a7">
+<div class="important" id="org09b426c">
 <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>
@@ -2152,7 +2146,7 @@ It just means that at some frequency, one of the closed-loop transfer functions
 </div>
 
 
-<div id="org6a4f956" class="figure">
+<div id="org4a1fd1a" class="figure">
 <p><img src="figs/results_sensitivity_hinf.png" alt="results_sensitivity_hinf.png" />
 </p>
 <p><span class="figure-number">Figure 29: </span>Weighting function and obtained closed-loop sensitivity</p>
@@ -2160,16 +2154,16 @@ It just means that at some frequency, one of the closed-loop transfer functions
 </div>
 </div>
 
-<div id="outline-container-org8ccbf6c" class="outline-3">
-<h3 id="org8ccbf6c"><span class="section-number-3">5.4</span> Shaping multiple closed-loop transfer functions</h3>
+<div id="outline-container-org5953b0e" class="outline-3">
+<h3 id="org5953b0e"><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="org7ad7325"></a>
+<a id="org32ad878"></a>
 </p>
 
 <p>
-As was shown in Section <a href="#orgb49c33d">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="#orgecd685d">7</a>.
+As was shown in Section <a href="#orgb088aa1">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="#org298b4d4">7</a>.
 </p>
 
 <p>
@@ -2177,7 +2171,7 @@ 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="#org9c4e967">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="#orgb0ce710">6</a>.
 </p>
 
 <p>
@@ -2185,14 +2179,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </p>
 <details><summary>Shaping of S and KS</summary>
 
-<div id="org19c7971" class="figure">
+<div id="org0e42d4c" 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 30: </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="#org19c7971">30</a></label><pre class="src src-matlab" id="org003a3b5">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="#org0e42d4c">30</a></label><pre class="src src-matlab" id="org65d94ea">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
      0   W2
      1  <span class="org-type">-</span>G];
 </pre>
@@ -2205,14 +2199,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </details>
 <details><summary>Shaping of S and T</summary>
 
-<div id="org870e7e4" class="figure">
+<div id="org05aace6" 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 31: </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="#org870e7e4">31</a></label><pre class="src src-matlab" id="org2c73c9f">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="#org05aace6">31</a></label><pre class="src src-matlab" id="orgbf61789">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>
@@ -2223,16 +2217,35 @@ Depending on the closed-loop transfer function being shaped, different general c
 <li>\(W_2\) is used to shape \(T\)</li>
 </ul>
 </details>
-<details><summary>Shaping of S, T and KS</summary>
+<details><summary>Shaping of S and GS</summary>
 
-<div id="orgc9741d7" class="figure">
-<p><img src="figs/general_conf_shaping_S_T_KS.png" alt="general_conf_shaping_S_T_KS.png" />
+<div id="org0b79570" 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 32: </span>Generalized Plant to shape \(S\), \(T\) and \(KS\)</p>
+<p><span class="figure-number">Figure 32: </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="#orgc9741d7">32</a></label><pre class="src src-matlab" id="org537febc">P = [W1 <span class="org-type">-</span>G<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="#org0b79570">32</a></label><pre class="src src-matlab" id="orgc317638">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>
+</div>
+<ul class="org-ul">
+<li>\(W_1\) is used to shape \(S\)</li>
+<li>\(W_2\) is used to shape \(GS\)</li>
+</ul>
+</details>
+<details><summary>Shaping of S, T and KS</summary>
+
+<div id="org74761d9" 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 33: </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="#org74761d9">33</a></label><pre class="src src-matlab" id="orgbcec50a">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];
@@ -2245,16 +2258,38 @@ Depending on the closed-loop transfer function being shaped, different general c
 <li>\(W_3\) is used to shape \(T\)</li>
 </ul>
 </details>
-<details><summary>Shaping of S, T, KS and GS</summary>
+<details><summary>Shaping of S, T and GS</summary>
 
-<div id="orgb3cf383" class="figure">
-<p><img src="figs/general_conf_shaping_S_T_KS_GS.png" alt="general_conf_shaping_S_T_KS_GS.png" />
+<div id="org72a1c6c" 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 \(S\), \(T\), \(KS\) and \(GS\)</p>
+<p><span class="figure-number">Figure 34: </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 5: </span>General Plant definition corresponding to Figure <a href="#orgb3cf383">33</a></label><pre class="src src-matlab" id="orgd2a76b6">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 6: </span>General Plant definition corresponding to Figure <a href="#org72a1c6c">34</a></label><pre class="src src-matlab" id="org92a3e5d">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];
+</pre>
+</div>
+
+<ul class="org-ul">
+<li>\(W_1\) is used to shape \(S\)</li>
+<li>\(W_2\) is used to shape \(GS\)</li>
+<li>\(W_3\) is used to shape \(T\)</li>
+</ul>
+</details>
+<details><summary>Shaping of S, T, KS and GS</summary>
+
+<div id="org053375d" 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 35: </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="#org053375d">35</a></label><pre class="src src-matlab" id="orgb33b0dc">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>
@@ -2278,7 +2313,7 @@ When shaping multiple closed-loop transfer functions, one should be verify caref
 
 
 
-<div class="warning" id="org392b255">
+<div class="warning" id="orgd60c6b6">
 <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:
@@ -2311,7 +2346,7 @@ 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="org2c01b09">
+<div class="exercice" id="org1eea0f0">
 <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>
@@ -2329,7 +2364,7 @@ You can follows this procedure for \(T\), \(KS\) and \(GS\):
 
 <details><summary>Answer</summary>
 <p>
-The obtained constrains are shown in Figure <a href="#orge022dd7">34</a>.
+The obtained constrains are shown in Figure <a href="#org54a8a92">36</a>.
 </p>
 </details>
 
@@ -2342,17 +2377,17 @@ 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="#orge022dd7">34</a>.
+These regions are summarized in Figure <a href="#org54a8a92">36</a>.
 </p>
 
 
-<div id="orge022dd7" class="figure">
+<div id="org54a8a92" 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 34: </span>Shaping the Gang of Four: Limitations</p>
+<p><span class="figure-number">Figure 36: </span>Shaping the Gang of Four: Limitations</p>
 </div>
 
-<div class="warning" id="orgf67ff3f">
+<div class="warning" id="orgd1cf553">
 <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.
@@ -2371,43 +2406,418 @@ Two approaches can be used to obtain controllers with reasonable order:
 </div>
 </div>
 
-<div id="outline-container-org3865b16" class="outline-2">
-<h2 id="org3865b16"><span class="section-number-2">6</span> Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example</h2>
+<div id="outline-container-org4cfe210" class="outline-2">
+<h2 id="org4cfe210"><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="org9c4e967"></a>
+<a id="orgb0ce710"></a>
 </p>
 </div>
 
-<div id="outline-container-org227c600" class="outline-3">
-<h3 id="org227c600"><span class="section-number-3">6.1</span> Problem</h3>
+<div id="outline-container-orgbb9bd9a" class="outline-3">
+<h3 id="orgbb9bd9a"><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>
+</ul>
+
+
+<ul class="org-ul">
+<li>Inputs Signals
+<ul class="org-ul">
+<li>Reference steps of 1mm</li>
+<li>Measurement noise is considered to be a white noise with a power spectral density of &#x2026;</li>
+</ul></li>
+<li>Specifications
+<ul class="org-ul">
+<li>Follow reference steps with a response time of 0.1s and static error less than \(1 \mu m\)</li>
+<li>Maximum control signal of 10N</li>
+<li>Robustness</li>
+<li>Reduce the effect of measurement noise on the position</li>
+</ul></li>
+</ul>
+
+
+<div class="org-src-container">
+<pre class="src src-matlab">k = 1e6; <span class="org-comment">% Stiffness [N/m]</span>
+c = 4e2; <span class="org-comment">% Damping [N/(m/s)]</span>
+m = 10; <span class="org-comment">% Mass [kg]</span>
+
+G = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c<span class="org-type">*</span>s <span class="org-type">+</span> k); <span class="org-comment">% Plant</span>
+Gd = (c<span class="org-type">*</span>s <span class="org-type">+</span> k)<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c<span class="org-type">*</span>s <span class="org-type">+</span> k); <span class="org-comment">% Disturbance</span>
+</pre>
 </div>
 
-<div id="outline-container-orge72fa1a" class="outline-3">
-<h3 id="orge72fa1a"><span class="section-number-3">6.2</span> Typical Procedure</h3>
-</div>
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-comment">% Generate Input Signals</span>
+t = 0<span class="org-type">:</span>1e<span class="org-type">-</span>3<span class="org-type">:</span>1;
 
-<div id="outline-container-org2c2c97a" class="outline-3">
-<h3 id="org2c2c97a"><span class="section-number-3">6.3</span> Step 1 - Shaping of the Sensitivity Function</h3>
-</div>
+r = zeros(size(t));
+r(t<span class="org-type">&gt;</span>0.1) = 1e<span class="org-type">-</span>3;
 
-<div id="outline-container-org0029581" class="outline-3">
-<h3 id="org0029581"><span class="section-number-3">6.4</span> Step 2 - Shaping of</h3>
+
+Fs = 1e3; <span class="org-comment">% Sampling Frequency [Hz]</span>
+Ts = 1<span class="org-type">/</span>Fs; <span class="org-comment">% Sampling Time [s]</span>
+n = sqrt(Fs<span class="org-type">/</span>2)<span class="org-type">*</span>randn(1, length(t)); <span class="org-comment">% Signal with an ASD equal to one</span>
+n = n<span class="org-type">*</span>1e<span class="org-type">-</span>6;
+
+d = zeros(size(t));
+d(t<span class="org-type">&gt;</span>0.5) = 5e<span class="org-type">-</span>4;
+</pre>
+</div>
 </div>
 </div>
 
-<div id="outline-container-org6864741" class="outline-2">
-<h2 id="org6864741"><span class="section-number-2">7</span> Conclusion</h2>
+<div id="outline-container-org7f64534" class="outline-3">
+<h3 id="org7f64534"><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">
+
+
+<colgroup>
+<col  class="org-left" />
+
+<col  class="org-left" />
+
+<col  class="org-left" />
+</colgroup>
+<thead>
+<tr>
+<th scope="col" class="org-left">&#xa0;</th>
+<th scope="col" class="org-left">&#xa0;</th>
+<th scope="col" class="org-left">&#xa0;</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="org-left">Response Time</td>
+<td class="org-left">&#xa0;</td>
+<td class="org-left">&#xa0;</td>
+</tr>
+
+<tr>
+<td class="org-left">Robustness</td>
+<td class="org-left">&#xa0;</td>
+<td class="org-left">&#xa0;</td>
+</tr>
+
+<tr>
+<td class="org-left">Limitation of the Command Amplitude</td>
+<td class="org-left">&#xa0;</td>
+<td class="org-left">&#xa0;</td>
+</tr>
+
+<tr>
+<td class="org-left">Filtering of the measurement noise</td>
+<td class="org-left">&#xa0;</td>
+<td class="org-left">&#xa0;</td>
+</tr>
+</tbody>
+</table>
+
+
+<div id="org9b19a55" class="figure">
+<p><img src="figs/mixed_sensitivity_control_schematic.png" alt="mixed_sensitivity_control_schematic.png" />
+</p>
+<p><span class="figure-number">Figure 37: </span>Generalized Plant used for the Mixed Sensitivity Synthesis</p>
+</div>
+
+<ul class="org-ul">
+<li>\(W_1\) is used to shape \(S\)</li>
+<li>\(W_2\) is used to shape \(KS\)</li>
+<li>\(W_3\) is used to shape \(T\)</li>
+</ul>
+
+<div class="org-src-container">
+<pre class="src src-matlab">P = [1 <span class="org-type">-</span>G
+     0  1
+     0  G
+     1 <span class="org-type">-</span>G];
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orga3ee604" class="outline-3">
+<h3 id="orga3ee604"><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="org4c8309d"></a>
+</p>
+
+<p>
+We start with the shaping of \(S\) alone.
+To not constrain \(KS\) and \(T\), we set small values for \(W_2\) and \(W_3\)
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">W2 = tf(1e<span class="org-type">-</span>8);
+W3 = tf(0.1);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">W1 = generateWeight(<span class="org-string">'G0'</span>, 1e3, ...
+                    <span class="org-string">'G1'</span>, 1<span class="org-type">/</span>2, ...
+                    <span class="org-string">'Gc'</span>, sqrt(2), <span class="org-string">'wc'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10, ...
+                    <span class="org-string">'n'</span>, 1);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Pw = blkdiag(W1, W2, W3, 1)<span class="org-type">*</span>P;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">K1 = hinfsyn(Pw, 1, 1, <span class="org-string">'Display'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<pre class="example" id="org1a4a18c">
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.5 &lt;=  gamma  &lt;=  0.552
+
+   gamma        X&gt;=0        Y&gt;=0       rho(XY)&lt;1    p/f
+  5.25e-01     0.0e+00     0.0e+00     6.061e-16     p
+  5.13e-01     0.0e+00     0.0e+00     0.000e+00     p
+  5.06e-01    -2.5e+00 #  -6.4e-14     1.440e-15     f
+  5.09e-01    -5.5e+00 #  -4.1e-14     9.510e-16     f
+  Limiting gains...
+  5.14e-01     0.0e+00     0.0e+00     1.039e-25     p
+  5.14e-01     0.0e+00     0.0e+00     1.040e-25     p
+
+  Best performance (actual): 0.514
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+</pre>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Z1 = lft(P, K1);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-comment">% Create model with r and n as inputs and y and u as outputs</span>
+Psim = [0  0  Gd  G
+        0  0  0   1
+        1 <span class="org-type">-</span>1 <span class="org-type">-</span>Gd <span class="org-type">-</span>G];
+Z1sim = lft(Psim, K1);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">z = lsim(Z1sim, [r;n;d], t);
+y1 = z(<span class="org-type">:</span>,1);
+u1 = z(<span class="org-type">:</span>,2);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-type">figure</span>;
+tiledlayout(1, 2, <span class="org-string">'TileSpacing'</span>, <span class="org-string">'None'</span>, <span class="org-string">'Padding'</span>, <span class="org-string">'None'</span>);
+
+nexttile;
+hold on;
+plot(t, y1);
+hold off;
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Output $y$ [m]'</span>);
+
+nexttile;
+hold on;
+plot(t, u1);
+hold off;
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Control Input $u$ [N]'</span>);
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org238cd32" class="outline-3">
+<h3 id="org238cd32"><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="orga5d1d24"></a>
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">W2 = generateWeight(<span class="org-string">'G0'</span>, 5e<span class="org-type">-</span>7, ...
+                    <span class="org-string">'G1'</span>, 1e<span class="org-type">-</span>3, ...
+                    <span class="org-string">'Gc'</span>, 1e<span class="org-type">-</span>6, <span class="org-string">'wc'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100, ...
+                    <span class="org-string">'n'</span>, 1);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Pw = blkdiag(W1, W2, W3, 1)<span class="org-type">*</span>P;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">K2 = hinfsyn(Pw, 1, 1, <span class="org-string">'Display'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<pre class="example" id="org245d1a9">
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.51 &lt;=  gamma  &lt;=  1.2
+
+   gamma        X&gt;=0        Y&gt;=0       rho(XY)&lt;1    p/f
+  7.82e-01     0.0e+00     0.0e+00     0.000e+00     p
+  6.32e-01    -2.2e+00 #  -3.0e-14     -1.136e-32     f
+  7.03e-01    -5.1e+00 #  -2.9e-14     5.128e-17     f
+  7.41e-01    -1.1e+01 #  -1.0e-14     1.431e-22     f
+  7.61e-01    -2.6e+01 #  -9.1e-15     1.215e-21     f
+  7.72e-01    -6.9e+01 #  -1.9e-14     2.828e-16     f
+  7.77e-01    -3.6e+02 #  -1.2e-14     1.213e-15     f
+
+  Best performance (actual): 0.782
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+</pre>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Z2 = lft(P, K2);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Z2sim = lft(Psim, K2);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">z = lsim(Z2sim, [r;n;d], t);
+y2 = z(<span class="org-type">:</span>,1);
+u2 = z(<span class="org-type">:</span>,2);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-type">figure</span>;
+tiledlayout(1, 2, <span class="org-string">'TileSpacing'</span>, <span class="org-string">'None'</span>, <span class="org-string">'Padding'</span>, <span class="org-string">'None'</span>);
+
+nexttile;
+hold on;
+plot(t, y1);
+plot(t, y2);
+hold off;
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Output $y$ [m]'</span>);
+
+nexttile;
+hold on;
+plot(t, u1);
+plot(t, u2);
+hold off;
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Control Input $u$ [N]'</span>);
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org98f0830" class="outline-3">
+<h3 id="org98f0830"><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="orge45b749"></a>
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">W3 = generateWeight(<span class="org-string">'G0'</span>, 1e<span class="org-type">-</span>1, ...
+                    <span class="org-string">'G1'</span>, 1e4, ...
+                    <span class="org-string">'Gc'</span>, 1, <span class="org-string">'wc'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30, ...
+                    <span class="org-string">'n'</span>, 3);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Pw = blkdiag(W1, W2, W3, 1)<span class="org-type">*</span>P;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">K3 = hinfsyn(Pw, 1, 1, <span class="org-string">'Display'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<pre class="example" id="org0c48d0d">
+K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.578 &lt;=  gamma  &lt;=  1.66
+
+   gamma        X&gt;=0        Y&gt;=0       rho(XY)&lt;1    p/f
+  9.78e-01    -1.3e+01 #  -6.2e-15     1.141e-18     f
+  1.27e+00     0.0e+00     0.0e+00     1.524e-15     p
+  1.12e+00     0.0e+00     0.0e+00     4.481e-15     p
+  1.04e+00    -4.0e+01 #  -1.0e-13     1.102e-42     f
+  1.08e+00    -6.6e+02 #  -4.4e-15     3.641e-18     f
+  1.10e+00     0.0e+00     0.0e+00     6.052e-21     p
+  1.09e+00     0.0e+00     0.0e+00     5.005e-15     p
+
+  Best performance (actual): 1.09
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+</pre>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Z3 = lft(P, K3);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Z3sim = lft(Psim, K3);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">z = lsim(Z3sim, [r;n;d], t);
+y3 = z(<span class="org-type">:</span>,1);
+u3 = z(<span class="org-type">:</span>,2);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-type">figure</span>;
+tiledlayout(1, 2, <span class="org-string">'TileSpacing'</span>, <span class="org-string">'None'</span>, <span class="org-string">'Padding'</span>, <span class="org-string">'None'</span>);
+
+nexttile;
+hold on;
+plot(t, y1);
+plot(t, y2);
+plot(t, y3);
+hold off;
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Output $y$ [m]'</span>);
+
+nexttile;
+hold on;
+plot(t, u1);
+plot(t, u2);
+plot(t, u3);
+hold off;
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Control Input $u$ [N]'</span>);
+</pre>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgfd49210" class="outline-2">
+<h2 id="orgfd49210"><span class="section-number-2">7</span> Conclusion</h2>
 <div class="outline-text-2" id="text-7">
 <p>
-<a id="org4814712"></a>
+<a id="org1b37b10"></a>
 </p>
 </div>
 </div>
 
-<div id="outline-container-org6094e8a" class="outline-2">
-<h2 id="org6094e8a">Resources</h2>
-<div class="outline-text-2" id="text-org6094e8a">
+<div id="outline-container-org4803826" class="outline-2">
+<h2 id="org4803826">Resources</h2>
+<div class="outline-text-2" id="text-org4803826">
 <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>
@@ -2420,7 +2830,7 @@ Two approaches can be used to obtain controllers with reasonable order:
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2020-12-01 mar. 18:49</p>
+<p class="date">Created: 2020-12-02 mer. 11:00</p>
 </div>
 </body>
 </html>
diff --git a/index.org b/index.org
index 9b37339..0867d17 100644
--- a/index.org
+++ b/index.org
@@ -41,15 +41,24 @@
 
 * Introduction                                                        :ignore:
 
-This document is structured as follows:
-- As $\mathcal{H}_\infty$ Control is a /model based/ control technique, a short introduction to model based control is given in Section [[sec:model_based_control]]
+The purpose of this document is to give a /practical introduction/ to the wonderful world of $\mathcal{H}_\infty$ Control.
+
+No attend is made to provide an exhaustive treatment of the subject.
+$\mathcal{H}_\infty$ Control is a very broad topic and entire books are written on it.
+Therefore, for more advanced discussion, please have a look at the recommended references at the bottom of this document.
+
+When possible, Matlab scripts used for the example/exercises are provided such that you can easily test them on your computer.
+
+
+The general structure of this document is as follows:
+- A short introduction to /model based control/ is given in Section [[sec:model_based_control]]
 - Classical /open/ loop shaping method is presented in Section [[sec:open_loop_shaping]].
-  It is also shown that $\mathcal{H}_\infty$ synthesis can be used for /open/ loop shaping.
-- $\mathcal{H}_\infty$
-  Important concepts such as the $\mathcal{H}_\infty$ norm and the generalized plant are introduced.
-- A
-- Finally, an complete example of the
-  is performed in Section [[sec:h_infinity_mixed_sensitivity]].
+  It is also shown that $\mathcal{H}_\infty$ synthesis can be used for /open/ loop shaping
+- Important concepts indispensable for $\mathcal{H}_\infty$ control such as the $\mathcal{H}_\infty$ norm and the generalized plant are introduced in Section [[sec:h_infinity_introduction]]
+- 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 [[sec:modern_interpretation_specification]]
+- 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 [[sec:closed-loop-shaping]]
+- Finally, complete examples of the use of $\mathcal{H}_\infty$ Control for practical problems are provided in Section [[sec:h_infinity_mixed_sensitivity]].
 
 * Matlab Init                                                :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -309,20 +318,6 @@ You can follow this generic procedure:
 \end{align}
 #+HTML: </details>
 #+end_exercice
-Hi Musa,
-Thank you very much for sharing this awesome package.
-For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by <details> blocks.
-
-For now, I am manually adding #+HTML: <details><summary>Code</summary> and #+HTML: </details> around the source blocks I want to hide...
-This is a very simple solution, but not so elegent nor practical.
-
-Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?
-
-
-Similarly, I would love to be able to export a <span> block with the name of the file corresponding to the source block.
-For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting!
-
-Thanks in advance
 
 Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure [[fig:mech_sys_1dof_inertial_contr]] into a classical feedback form as shown in Figure [[fig:open_loop_shaping]].
 
@@ -947,7 +942,7 @@ where:
 
 The procedure to convert a typical control architecture as the one shown in Figure [[fig:classical_feedback_tracking]] to a generalized Plant is as follows:
 1. Define signals ($w$, $z$, $u$ and $v$) of the generalized plant
-2. Remove $K$ and rearrange the inputs and outputs to match the generalized configuration
+2. Remove $K$ and rearrange the inputs and outputs to match the generalized configuration shown in Figure [[fig:general_plant]]
 
 #+begin_src latex :file classical_feedback_tracking.pdf
   \begin{tikzpicture}
@@ -994,7 +989,8 @@ The procedure to convert a typical control architecture as the one shown in Figu
 #+end_src
 
 #+begin_exercice
-  Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
+  1. Convert the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]] to a generalized configuration
+  2. Compute the transfer function matrix using Matlab as a function or $K$ and $G$
 
   #+name: fig:classical_feedback_tracking
   #+caption: Classical Feedback Control Architecture (Tracking)
@@ -1003,11 +999,12 @@ The procedure to convert a typical control architecture as the one shown in Figu
 #+HTML: <details><summary>Hint</summary>
   First, define the signals of the generalized plant:
   - Exogenous inputs: $w = r$
-  - Signals to be minimized: $z_1 = \epsilon$, $z_2 = u$
-  - Control signals: $v = y$
-  - Control inputs: $u$
+  - Signals to be minimized:
+    Usually, we want to minimize the tracking errors $\epsilon$ and the control signal $u$: $z = [\epsilon,\ u]$
+  - Controller inputs: this is the signal at the input of the controller: $v = \epsilon$
+  - Control inputs: signal generated by the controller: $u$
 
-  Then, Remove $K$ and rearrange the inputs and outputs.
+  Then, Remove $K$ and rearrange the inputs and outputs as in Figure [[fig:general_plant]].
 #+HTML: </details>
 
 #+HTML: <details><summary>Answer</summary>
@@ -1016,18 +1013,18 @@ The procedure to convert a typical control architecture as the one shown in Figu
   #+name: fig:mixed_sensitivity_ref_tracking
   #+caption: Generalized plant of the Classical Feedback Control Architecture (Tracking)
   [[file:figs/mixed_sensitivity_ref_tracking.png]]
+
+  Using Matlab, the generalized plant can be defined as follows:
+  #+begin_src matlab :tangle no :eval no
+    P = [1 -G;
+         0  1;
+         1 -G]
+    P.InputName = {'w', 'u'};
+    P.OutputName = {'e', 'u', 'v'};
+  #+end_src
 #+HTML: </details>
 #+end_exercice
 
-Using Matlab, the generalized plant can be defined as follows:
-#+begin_src matlab :tangle no :eval no
-  P = [1 -G;
-       0  1;
-       1 -G]
-  P.InputName = {'w', 'u'};
-  P.OutputName = {'e', 'u', 'v'};
-#+end_src
-
 * Modern Interpretation of Control Specifications
 <<sec:modern_interpretation_specification>>
 
@@ -2087,6 +2084,58 @@ Depending on the closed-loop transfer function being shaped, different general c
 - $W_2$ is used to shape $T$
 #+HTML: </details>
 
+*** S GS                                                            :ignore:
+#+HTML: <details><summary>Shaping of S and GS</summary>
+#+begin_src latex :file general_conf_shaping_S_GS.pdf
+  \begin{tikzpicture}
+    % Blocs
+    \node[block] (G) {$G$};
+
+    \node[addb={+}{-}{}{}{}, left=0.8 of G] (addw) {};
+    \node[block, above right=0.4 and 0.8 of G] (W2) {$W_2$};
+    \node[block, above=0.5 of W2] (W1) {$W_1$};
+
+    \begin{scope}[on background layer]
+      \node[fit={(addw.west |- G.south) (W1.north east)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
+      \node[above] at (P.north) {Weighted Generalized Plant $P$};
+    \end{scope}
+
+    \node[block, below=0.6 of P] (K) {$K$};
+
+    \coordinate[right=1.0 of W1] (z);
+    \coordinate[above left=2.4 and 1.0 of addw] (w);
+
+    % Connections
+    \draw[->] (addw.east) -- (G.west);
+    \draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
+    \draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W2.west);
+
+    \draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
+    \draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
+
+    \draw[->] (G.east) -- (G-|z) |- node[near start, right]{$v$} (K.east);
+    \draw[->] (K.west) -| node[near end, left]{$u$} (addw-|w) -- (addw.west);
+
+    \draw[->] (w) node[above right]{$w$} -| (addw.north);
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:general_conf_shaping_S_GS
+#+caption: Generalized Plant to shape $S$ and $GS$
+#+RESULTS:
+[[file:figs/general_conf_shaping_S_GS.png]]
+
+#+name: lst:general_plant_S_GS
+#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_GS]]
+#+begin_src matlab :eval no :tangle no
+  P = [W1   -W1
+       G*W2 -G*W2
+       G    -G];
+#+end_src
+- $W_1$ is used to shape $S$
+- $W_2$ is used to shape $GS$
+#+HTML: </details>
+
 *** S T KS                                                          :ignore:
 #+HTML: <details><summary>Shaping of S, T and KS</summary>
 #+begin_src latex :file general_conf_shaping_S_T_KS.pdf
@@ -2147,6 +2196,66 @@ Depending on the closed-loop transfer function being shaped, different general c
 - $W_3$ is used to shape $T$
 #+HTML: </details>
 
+*** S T GS                                                          :ignore:
+#+HTML: <details><summary>Shaping of S, T and GS</summary>
+#+begin_src latex :file general_conf_shaping_S_T_GS.pdf
+  \begin{tikzpicture}
+    % Blocs
+    \node[block] (G) {$G$};
+
+    \node[addb={+}{-}{}{}{}, left=0.8 of G] (addw) {};
+    \node[block, above right=0.4 and 0.8 of G] (W3) {$W_3$};
+    \node[block, above=0.2 of W3] (W2) {$W_2$};
+    \node[block, above=0.2 of W2] (W1) {$W_1$};
+
+    \coordinate (addwin) at ($(addw.west)+(-0.5, 0)$);
+
+    \begin{scope}[on background layer]
+      \node[fit={(addwin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
+      \node[above] at (P.north) {Weighted Generalized Plant $P$};
+    \end{scope}
+
+    \node[block, below=0.6 of P] (K) {$K$};
+
+    \coordinate[right=0.8 of W1] (z);
+    \coordinate[above left=2.4 and 0.8 of addwin] (w);
+
+    % Connections
+    \draw[->] (addw.east) -- (G.west);
+    \draw[->] ($(addw.east)+( 0.3, 0)$)node[branch]{} |- (W1.west);
+    \draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W2.west);
+    \draw[->] (addwin)node[branch]{} |- (W3.west);
+
+    \draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
+    \draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
+    \draw[->] (W3.east) -- (W3-|z) node[above left](z3){$z_3$};
+
+    \draw[->] (G.east) -- (G-|z) |- node[near start, right]{$v$} (K.east);
+    \draw[->] (K.west) -| node[near end, left]{$u$} (addw-|w) -- (addw.west);
+
+    \draw[->] (w) node[above right]{$w$} -| (addw.north);
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:general_conf_shaping_S_T_GS
+#+caption: Generalized Plant to shape $S$, $T$ and $GS$
+#+RESULTS:
+[[file:figs/general_conf_shaping_S_T_GS.png]]
+
+#+name: lst:general_plant_S_T_GS
+#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_GS]]
+#+begin_src matlab :eval no :tangle no
+  P = [W1   -W1
+       G*W2 -G*W2
+       0     W3
+       G    -G];
+#+end_src
+
+- $W_1$ is used to shape $S$
+- $W_2$ is used to shape $GS$
+- $W_3$ is used to shape $T$
+#+HTML: </details>
+
 *** S T KS GS                                                       :ignore:
 #+HTML: <details><summary>Shaping of S, T, KS and GS</summary>
 #+begin_src latex :file general_conf_shaping_S_T_KS_GS.pdf
@@ -2341,13 +2450,1409 @@ These regions are summarized in Figure [[fig:h-infinity-4-blocs-constrains]].
 * Mixed-Sensitivity $\mathcal{H}_\infty$ Control - Example
 <<sec:h_infinity_mixed_sensitivity>>
 
-** Problem
+** Control Problem
 
-** Typical Procedure
+- [ ] Control Diagram
 
-** Step 1 - Shaping of the Sensitivity Function
 
-** Step 2 - Shaping of
+- Inputs Signals
+  - Reference steps of 1mm
+  - Measurement noise is considered to be a white noise with a power spectral density of ...
+- Specifications
+  - Follow reference steps with a response time of 0.1s and static error less than $1 \mu m$
+  - Maximum control signal of 10N
+  - Robustness
+  - Reduce the effect of measurement noise on the position
+
+
+#+begin_src matlab
+  k = 1e6; % Stiffness [N/m]
+  c = 4e2; % Damping [N/(m/s)]
+  m = 10; % Mass [kg]
+
+  G = 1/(m*s^2 + c*s + k); % Plant
+  Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
+#+end_src
+
+#+begin_src matlab
+  % Generate Input Signals
+  t = 0:1e-3:1;
+
+  r = zeros(size(t));
+  r(t>0.1) = 1e-3;
+
+
+  Fs = 1e3; % Sampling Frequency [Hz]
+  Ts = 1/Fs; % Sampling Time [s]
+  n = sqrt(Fs/2)*randn(1, length(t)); % Signal with an ASD equal to one
+  n = n*1e-6;
+
+  d = zeros(size(t));
+  d(t>0.5) = 5e-4;
+#+end_src
+
+** Control Design Procedure
+
+|                                     |   |   |
+|-------------------------------------+---+---|
+| Response Time                       |   |   |
+| Robustness                          |   |   |
+| Limitation of the Command Amplitude |   |   |
+| Filtering of the measurement noise  |   |   |
+
+#+begin_src latex :file mixed_sensitivity_control_schematic.pdf
+  \begin{tikzpicture}
+    % Blocs
+    \node[block] (G) {$G$};
+
+    \node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {};
+    \node[block, above right=0.4 and 0.8 of addw] (W3) {$W_3$};
+    \node[block, above=0.2 of W3] (W2) {$W_2$};
+    \node[block, above=0.2 of W2] (W1) {$W_1$};
+
+    \coordinate (Gin) at ($(G.west)+(-0.5, 0)$);
+
+    \begin{scope}[on background layer]
+      \node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
+      \node[above] at (P.north) {Weighted Generalized Plant $P$};
+    \end{scope}
+
+    \node[block, below=0.6 of P] (K) {$K$};
+
+    \coordinate[right=0.8 of W1] (z);
+    \coordinate[above left=1.4 and 1.3 of G] (w);
+
+    % Connections
+    \draw[->] (G.east) -- (addw.west);
+    \draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
+    \draw[->] (Gin)node[branch]{} |- (W2.west);
+    \draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W3.west);
+
+    \draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
+    \draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
+    \draw[->] (W3.east) -- (W3-|z) node[above left](z3){$z_3$};
+
+    \draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
+    \draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
+
+    \draw[->] (w) node[above right]{$w$} -| (addw.north);
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:mixed_sensitivity_control_schematic
+#+caption: Generalized Plant used for the Mixed Sensitivity Synthesis
+#+RESULTS:
+[[file:figs/mixed_sensitivity_control_schematic.png]]
+
+- $W_1$ is used to shape $S$
+- $W_2$ is used to shape $KS$
+- $W_3$ is used to shape $T$
+
+#+begin_src matlab
+  P = [1 -G
+       0  1
+       0  G
+       1 -G];
+#+end_src
+
+** Step 1 - Shaping of $S$
+<<sec:step_shaping_S>>
+
+We start with the shaping of $S$ alone.
+To not constrain $KS$ and $T$, we set small values for $W_2$ and $W_3$
+
+#+begin_src matlab
+  W2 = tf(1e-8);
+  W3 = tf(0.1);
+#+end_src
+
+#+begin_src matlab
+  W1 = generateWeight('G0', 1e3, ...
+                      'G1', 1/2, ...
+                      'Gc', sqrt(2), 'wc', 2*pi*10, ...
+                      'n', 1);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.5 <=  gamma  <=  0.552
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  5.25e-01     0.0e+00     0.0e+00     6.061e-16     p
+  5.13e-01     0.0e+00     0.0e+00     0.000e+00     p
+  5.06e-01    -2.5e+00 #  -6.4e-14     1.440e-15     f
+  5.09e-01    -5.5e+00 #  -4.1e-14     9.510e-16     f
+  Limiting gains...
+  5.14e-01     0.0e+00     0.0e+00     1.039e-25     p
+  5.14e-01     0.0e+00     0.0e+00     1.040e-25     p
+
+  Best performance (actual): 0.514
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z1 = lft(P, K1);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 4, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('KS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  % Create model with r and n as inputs and y and u as outputs
+  Psim = [0  0  Gd  G
+          0  0  0   1
+          1 -1 -Gd -G];
+  Z1sim = lft(Psim, K1);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z1sim, [r;n;d], t);
+  y1 = z(:,1);
+  u1 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Step 2 - Shaping of $KS$
+<<sec:step_shaping_KS>>
+
+#+begin_src matlab
+  W2 = generateWeight('G0', 5e-7, ...
+                      'G1', 1e-3, ...
+                      'Gc', 1e-6, 'wc', 2*pi*100, ...
+                      'n', 1);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K2 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.51 <=  gamma  <=  1.2
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  7.82e-01     0.0e+00     0.0e+00     0.000e+00     p
+  6.32e-01    -2.2e+00 #  -3.0e-14     -1.136e-32     f
+  7.03e-01    -5.1e+00 #  -2.9e-14     5.128e-17     f
+  7.41e-01    -1.1e+01 #  -1.0e-14     1.431e-22     f
+  7.61e-01    -2.6e+01 #  -9.1e-15     1.215e-21     f
+  7.72e-01    -6.9e+01 #  -1.9e-14     2.828e-16     f
+  7.77e-01    -3.6e+02 #  -1.2e-14     1.213e-15     f
+
+  Best performance (actual): 0.782
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z2 = lft(P, K2);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('KS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  Z2sim = lft(Psim, K2);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z2sim, [r;n;d], t);
+  y2 = z(:,1);
+  u2 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, y2);
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  plot(t, u2);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Step 3 - Shaping of $T$
+<<sec:step_shaping_T>>
+
+#+begin_src matlab
+  W3 = generateWeight('G0', 1e-1, ...
+                      'G1', 1e4, ...
+                      'Gc', 1, 'wc', 2*pi*30, ...
+                      'n', 3);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.578 <=  gamma  <=  1.66
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  9.78e-01    -1.3e+01 #  -6.2e-15     1.141e-18     f
+  1.27e+00     0.0e+00     0.0e+00     1.524e-15     p
+  1.12e+00     0.0e+00     0.0e+00     4.481e-15     p
+  1.04e+00    -4.0e+01 #  -1.0e-13     1.102e-42     f
+  1.08e+00    -6.6e+02 #  -4.4e-15     3.641e-18     f
+  1.10e+00     0.0e+00     0.0e+00     6.052e-21     p
+  1.09e+00     0.0e+00     0.0e+00     5.005e-15     p
+
+  Best performance (actual): 1.09
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z3 = lft(P, K3);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('KS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  Z3sim = lft(Psim, K3);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z3sim, [r;n;d], t);
+  y3 = z(:,1);
+  u3 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, y2);
+  plot(t, y3);
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  plot(t, u2);
+  plot(t, u3);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+* Alternative Example - Shaping of $GS$                             :noexport:
+** Plant
+#+begin_src matlab
+  k = 1e6; % Stiffness [N/m]
+  c = 4e2; % Damping [N/(m/s)]
+  m = 10; % Mass [kg]
+
+  G = 1/(m*s^2 + c*s + k); % Plant
+  Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
+#+end_src
+
+#+begin_src matlab
+  % Generate Input Signals
+  t = 0:1e-4:0.5;
+
+  r = zeros(size(t));
+  r(t>0.1) = 1e-3;
+
+
+  Fs = 1e3; % Sampling Frequency [Hz]
+  Ts = 1/Fs; % Sampling Time [s]
+  n = sqrt(Fs/2)*randn(1, length(t)); % Signal with an ASD equal to one
+  n = n*1e-5;
+
+  d = zeros(size(t));
+  d(t>0.25) = 5e-4;
+#+end_src
+
+#+begin_src matlab
+  P = [1 -1
+       G -G
+       0  1
+       G -G];
+#+end_src
+
+** Step 1 - Shaping of $S$
+#+begin_src matlab
+  W1 = generateWeight('G0', 1e3, ...
+                      'G1', 1/2, ...
+                      'Gc', 1, 'wc', 2*pi*10, ...
+                      'n', 1);
+#+end_src
+
+#+begin_src matlab
+  W2 = tf(1e-8);
+  W3 = tf(1e-8);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  18.5 <=  gamma  <=  18.9
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  1.87e+01     0.0e+00     0.0e+00     5.225e+01     p
+
+  Best performance (actual): 0.252
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z1 = lft(P, K1);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('GS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  % Create model with r and n as inputs and y and u as outputs
+  Psim = [0  0  Gd  G
+          0  0  0   1
+          1 -1 -Gd -G];
+  Z1sim = lft(Psim, K1);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z1sim, [r;n;d], t);
+  y1 = z(:,1);
+  u1 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Step 2 - Shaping of $GS$
+<<sec:step_shaping_GS>>
+
+#+begin_src matlab
+  W2 = generateWeight('G0', 3e5, ...
+                      'G1', 1e9, ...
+                      'Gc', 1e6, 'wc', 2*pi*200, ...
+                      'n', 1);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K2 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.51 <=  gamma  <=  1.2
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  7.82e-01     0.0e+00     0.0e+00     0.000e+00     p
+  6.32e-01    -2.2e+00 #  -3.0e-14     -1.136e-32     f
+  7.03e-01    -5.1e+00 #  -2.9e-14     5.128e-17     f
+  7.41e-01    -1.1e+01 #  -1.0e-14     1.431e-22     f
+  7.61e-01    -2.6e+01 #  -9.1e-15     1.215e-21     f
+  7.72e-01    -6.9e+01 #  -1.9e-14     2.828e-16     f
+  7.77e-01    -3.6e+02 #  -1.2e-14     1.213e-15     f
+
+  Best performance (actual): 0.782
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z2 = lft(P, K2);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('GS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  Z2sim = lft(Psim, K2);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z2sim, [r;n;d], t);
+  y2 = z(:,1);
+  u2 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, y2);
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  plot(t, u2);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Step 3 - Shaping of $T$
+<<sec:step_shaping_T>>
+
+#+begin_src matlab
+  W3 = generateWeight('G0', 1e-1, ...
+                      'G1', 1e4, ...
+                      'Gc', 1, 'wc', 2*pi*80, ...
+                      'n', 2);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.578 <=  gamma  <=  1.66
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  9.78e-01    -1.3e+01 #  -6.2e-15     1.141e-18     f
+  1.27e+00     0.0e+00     0.0e+00     1.524e-15     p
+  1.12e+00     0.0e+00     0.0e+00     4.481e-15     p
+  1.04e+00    -4.0e+01 #  -1.0e-13     1.102e-42     f
+  1.08e+00    -6.6e+02 #  -4.4e-15     3.641e-18     f
+  1.10e+00     0.0e+00     0.0e+00     6.052e-21     p
+  1.09e+00     0.0e+00     0.0e+00     5.005e-15     p
+
+  Best performance (actual): 1.09
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z3 = lft(P, K3);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('GS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  Z3sim = lft(Psim, K3);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z3sim, [r;n;d], t);
+  y3 = z(:,1);
+  u3 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, y2);
+  plot(t, y3);
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  plot(t, u2);
+  plot(t, u3);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Evolution of Controller
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 4, 1000);
+
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile([2,1]);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(K3, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+  hold off;
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(K1, freqs, 'Hz')))));
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(K2, freqs, 'Hz')))));
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(K3, freqs, 'Hz')))));
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  yticks(-360:90:360); ylim([-270, 90]);
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  hold off;
+  linkaxes([ax1,ax2],'x');
+  xlim([freqs(1), freqs(end)]);
+#+end_src
+
+Loop Gain
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 4, 1000);
+
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile([2,1]);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G*K1, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G*K2, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G*K3, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+  hold off;
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G*K1, freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G*K2, freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G*K3, freqs, 'Hz'))));
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  yticks(-360:90:360); ylim([-270, 90]);
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  hold off;
+  linkaxes([ax1,ax2],'x');
+  xlim([freqs(1), freqs(end)]);
+#+end_src
+
+* Alternative Example - Follow Triangle Reference                   :noexport:
+** Plant
+#+begin_src matlab
+  k = 1e6; % Stiffness [N/m]
+  c = 4e2; % Damping [N/(m/s)]
+  m = 10; % Mass [kg]
+
+  G = 1/(m*s^2 + c*s + k); % Plant
+  Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
+#+end_src
+
+#+begin_src matlab
+  % Generate Input Signals
+  t = 0:1e-4:0.5;
+
+  r = zeros(size(t));
+  r(t>0.1 & t<=0.2) = 1e-2*(t(t>0.1 & t<=0.2)-0.1);
+  r(t>0.2) = 1e-3;
+
+
+  Fs = 1e3; % Sampling Frequency [Hz]
+  Ts = 1/Fs; % Sampling Time [s]
+  n = sqrt(Fs/2)*randn(1, length(t)); % Signal with an ASD equal to one
+  n = n*1e-6;
+
+  d = zeros(size(t));
+  d(t>0.3) = 5e-4;
+#+end_src
+
+#+begin_src matlab
+  P = [1 -1
+       G -G
+       0  1
+       G -G];
+#+end_src
+
+** Step 1 - Shaping of $S$
+#+begin_src matlab
+  W1 = generateWeight('G0', 1e3, ...
+                      'G1', 1/2, ...
+                      'Gc', 1, 'wc', 2*pi*10, ...
+                      'n', 1);
+#+end_src
+
+#+begin_src matlab
+  W2 = tf(1e-8);
+  W3 = tf(1e-8);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  18.5 <=  gamma  <=  18.9
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  1.87e+01     0.0e+00     0.0e+00     5.225e+01     p
+
+  Best performance (actual): 0.252
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z1 = lft(P, K1);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('GS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  % Create model with r and n as inputs and y and u as outputs
+  Psim = [0  0  Gd  G
+          0  0  0   1
+          1 -1 -Gd -G];
+  Z1sim = lft(Psim, K1);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z1sim, [r;n;d], t);
+  y1 = z(:,1);
+  u1 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, r, 'k--');
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Step 1 - Shaping of $S$ with 2 integrators
+#+begin_src matlab
+  W1 = generateWeight('G0', 1e3, ...
+                      'G1', 1/2, ...
+                      'Gc', 1, 'wc', 2*pi*15, ...
+                      'n', 2);
+#+end_src
+
+#+begin_src matlab
+  W2 = tf(1e-8);
+  W3 = tf(1e-8);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K1b = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  18.5 <=  gamma  <=  18.9
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  1.87e+01     0.0e+00     0.0e+00     5.225e+01     p
+
+  Best performance (actual): 0.252
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z1b = lft(P, K1b);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z1b(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z1b(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('GS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z1b(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  % Create model with r and n as inputs and y and u as outputs
+  Psim = [0  0  Gd  G
+          0  0  0   1
+          1 -1 -Gd -G];
+  Z1bsim = lft(Psim, K1b);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z1bsim, [r;n;d], t);
+  y1b = z(:,1);
+  u1b = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, y1b);
+  plot(t, r, 'k--');
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  plot(t, u1b);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Step 2 - Shaping of $GS$
+<<sec:step_shaping_GS>>
+
+#+begin_src matlab
+  W2 = tf(4e5);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K2 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.51 <=  gamma  <=  1.2
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  7.82e-01     0.0e+00     0.0e+00     0.000e+00     p
+  6.32e-01    -2.2e+00 #  -3.0e-14     -1.136e-32     f
+  7.03e-01    -5.1e+00 #  -2.9e-14     5.128e-17     f
+  7.41e-01    -1.1e+01 #  -1.0e-14     1.431e-22     f
+  7.61e-01    -2.6e+01 #  -9.1e-15     1.215e-21     f
+  7.72e-01    -6.9e+01 #  -1.9e-14     2.828e-16     f
+  7.77e-01    -3.6e+02 #  -1.2e-14     1.213e-15     f
+
+  Best performance (actual): 0.782
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z2 = lft(P, K2);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('GS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  Z2sim = lft(Psim, K2);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z2sim, [r;n;d], t);
+  y2 = z(:,1);
+  u2 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, y2);
+  plot(t, r, 'k--');
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  plot(t, u2);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Step 3 - Shaping of $T$
+<<sec:step_shaping_T>>
+
+#+begin_src matlab
+  W3 = generateWeight('G0', 1e-1, ...
+                      'G1', 1e4, ...
+                      'Gc', 1, 'wc', 2*pi*70, ...
+                      'n', 2);
+#+end_src
+
+#+begin_src matlab
+  Pw = blkdiag(W1, W2, W3, 1)*P;
+#+end_src
+
+#+begin_src matlab :results output replace
+  K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  Test bounds:  0.578 <=  gamma  <=  1.66
+
+   gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  9.78e-01    -1.3e+01 #  -6.2e-15     1.141e-18     f
+  1.27e+00     0.0e+00     0.0e+00     1.524e-15     p
+  1.12e+00     0.0e+00     0.0e+00     4.481e-15     p
+  1.04e+00    -4.0e+01 #  -1.0e-13     1.102e-42     f
+  1.08e+00    -6.6e+02 #  -4.4e-15     3.641e-18     f
+  1.10e+00     0.0e+00     0.0e+00     6.052e-21     p
+  1.09e+00     0.0e+00     0.0e+00     5.005e-15     p
+
+  Best performance (actual): 1.09
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+#+end_example
+
+#+begin_src matlab
+  Z3 = lft(P, K3);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 3, 1000);
+
+  figure;
+  tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(1,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('S');
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(2,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('GS');
+
+  ax3 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Z3(3,1), freqs, 'Hz'))));
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  title('T');
+#+end_src
+
+#+begin_src matlab
+  Z3sim = lft(Psim, K3);
+#+end_src
+
+#+begin_src matlab
+  z = lsim(Z3sim, [r;n;d], t);
+  y3 = z(:,1);
+  u3 = z(:,2);
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  nexttile;
+  hold on;
+  plot(t, y1);
+  plot(t, y2);
+  plot(t, y3);
+  plot(t, r, 'k--');
+  hold off;
+  xlabel('Time [s]'); ylabel('Output $y$ [m]');
+
+  nexttile;
+  hold on;
+  plot(t, u1);
+  plot(t, u2);
+  plot(t, u3);
+  hold off;
+  xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
+#+end_src
+
+** Evolution of Controller
+
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 4, 1000);
+
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile([2,1]);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(K3, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+  hold off;
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(K1, freqs, 'Hz')))));
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(K2, freqs, 'Hz')))));
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(K3, freqs, 'Hz')))));
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  yticks(-360:90:360); ylim([-270, 90]);
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  hold off;
+  linkaxes([ax1,ax2],'x');
+  xlim([freqs(1), freqs(end)]);
+#+end_src
+
+Loop Gain
+#+begin_src matlab :exports none
+  freqs = logspace(-1, 4, 1000);
+
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile([2,1]);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G*K1, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G*K2, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G*K3, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+  hold off;
+
+  ax2 = nexttile;
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G*K1, freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G*K2, freqs, 'Hz'))));
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G*K3, freqs, 'Hz'))));
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  yticks(-360:90:360); ylim([-270, 90]);
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  hold off;
+  linkaxes([ax1,ax2],'x');
+  xlim([freqs(1), freqs(end)]);
+#+end_src
 
 * Conclusion
 <<sec:conclusion>>