Correct hinf notation
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-11-25 mer. 19:37 -->
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<!-- 2020-11-25 mer. 19:38 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Robust Control - \(\mathcal{H}_\infty\) Synthesis</title>
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<meta name="generator" content="Org mode" />
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@ -30,29 +30,29 @@
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org4643b2e">1. Introduction to the Control Methodology - Model Based Control</a></li>
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<li><a href="#org8a683c3">2. Some Background: From Classical Control to Robust Control</a></li>
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<li><a href="#orgc683003">3. The \(\mathcal{H}_\infty\) Norm</a></li>
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<li><a href="#org23e7e4a">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#orgcd078d9">5. The Generalized Plant</a></li>
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<li><a href="#org675f3d8">6. Problem Formulation</a></li>
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<li><a href="#org64eaa6d">7. Classical feedback control and closed loop transfer functions</a></li>
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<li><a href="#orgee3fbb9">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
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<li><a href="#orgc9be7b5">9. Modern Interpretation of the Control Specifications</a>
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<li><a href="#org983c7b8">1. Introduction to the Control Methodology - Model Based Control</a></li>
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<li><a href="#org56e9b1e">2. Some Background: From Classical Control to Robust Control</a></li>
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<li><a href="#org26e4f77">3. The \(\mathcal{H}_\infty\) Norm</a></li>
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<li><a href="#org7fece23">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#orga8463c6">5. The Generalized Plant</a></li>
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<li><a href="#org8f2f474">6. Problem Formulation</a></li>
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<li><a href="#org6e6fa08">7. Classical feedback control and closed loop transfer functions</a></li>
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<li><a href="#org61c8d78">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
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<li><a href="#orgf0b775f">9. Modern Interpretation of the Control Specifications</a>
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<ul>
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<li><a href="#orgf9aaff9">9.1. Introduction</a></li>
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<li><a href="#org0691a02">9.1. Introduction</a></li>
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</ul>
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</li>
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<li><a href="#org42f3566">10. Resources</a></li>
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<li><a href="#org2d9c766">10. Resources</a></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org4643b2e" class="outline-2">
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<h2 id="org4643b2e"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2>
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<div id="outline-container-org983c7b8" class="outline-2">
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<h2 id="org983c7b8"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org3f1eee3">1</a>.
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The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org893c4a9">1</a>.
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It consists of three steps:
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</p>
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<ol class="org-ol">
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@ -66,7 +66,7 @@ It consists of three steps:
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</ol>
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<div id="org3f1eee3" class="figure">
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<div id="org893c4a9" class="figure">
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<p><img src="figs/control-procedure.png" alt="control-procedure.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Typical Methodoly for Model Based Control</p>
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@ -78,8 +78,8 @@ In this document, we will mainly focus on steps 2 and 3.
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</div>
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</div>
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<div id="outline-container-org8a683c3" class="outline-2">
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<h2 id="org8a683c3"><span class="section-number-2">2</span> Some Background: From Classical Control to Robust Control</h2>
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<div id="outline-container-org56e9b1e" class="outline-2">
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<h2 id="org56e9b1e"><span class="section-number-2">2</span> Some Background: From Classical Control to Robust Control</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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Classical Control (1930)
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@ -156,10 +156,10 @@ Robust Control (1980)
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</div>
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</div>
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<div id="outline-container-orgc683003" class="outline-2">
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<h2 id="orgc683003"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2>
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<div id="outline-container-org26e4f77" class="outline-2">
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<h2 id="org26e4f77"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2>
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<div class="outline-text-2" id="text-3">
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<div class="definition" id="org376138c">
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<div class="definition" id="org2a3b6b9">
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<p>
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The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
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</p>
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@ -176,7 +176,7 @@ For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a
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</div>
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<div class="exampl" id="org4a6bc14">
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<div class="exampl" id="org2014425">
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<p>
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Let’s define a plant dynamics:
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</p>
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@ -201,12 +201,12 @@ And compute its \(\mathcal{H}_\infty\) norm using the <code>hinfnorm</code> func
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<p>
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The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure <a href="#org109ee6e">2</a>.
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The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure <a href="#org614e629">2</a>.
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The maximum value of the magnitude over all frequencies does correspond to the \(\mathcal{H}_\infty\) norm of \(G(s)\) as Equation \eqref{eq:hinf_norm_siso} implies.
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</p>
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<div id="org109ee6e" class="figure">
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<div id="org614e629" class="figure">
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<p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p>
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@ -216,46 +216,46 @@ The maximum value of the magnitude over all frequencies does correspond to the \
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</div>
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</div>
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<div id="outline-container-org23e7e4a" class="outline-2">
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<h2 id="org23e7e4a"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2>
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<div id="outline-container-org7fece23" class="outline-2">
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<h2 id="org7fece23"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2>
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<div class="outline-text-2" id="text-4">
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<p>
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<b>Optimization problem</b>:
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\(\hinf\) synthesis is a method that uses an <b>algorithm</b> (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the \(\hinf\) norms of defined transfer functions are minimized.
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\(\mathcal{H}_\infty\) synthesis is a method that uses an <b>algorithm</b> (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the \(\mathcal{H}_\infty\) norms of defined transfer functions are minimized.
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</p>
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<p>
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<b>Engineer work</b>:
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</p>
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<ol class="org-ol">
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<li>Write the problem as standard \(\hinf\) problem</li>
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<li>Translate the specifications as \(\hinf\) norms</li>
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<li>Write the problem as standard \(\mathcal{H}_\infty\) problem</li>
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<li>Translate the specifications as \(\mathcal{H}_\infty\) norms</li>
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<li>Make the synthesis and analyze the obtain controller</li>
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<li>Reduce the order of the controller for implementation</li>
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</ol>
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<p>
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<b>Many ways to use the \(\hinf\) Synthesis</b>:
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<b>Many ways to use the \(\mathcal{H}_\infty\) Synthesis</b>:
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</p>
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<ul class="org-ul">
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<li>Traditional \(\hinf\) Synthesis</li>
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<li>Traditional \(\mathcal{H}_\infty\) Synthesis</li>
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<li>Mixed Sensitivity Loop Shaping</li>
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<li>Fixed-Structure \(\hinf\) Synthesis</li>
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<li>Signal Based \(\hinf\) Synthesis</li>
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<li>Fixed-Structure \(\mathcal{H}_\infty\) Synthesis</li>
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<li>Signal Based \(\mathcal{H}_\infty\) Synthesis</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgcd078d9" class="outline-2">
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<h2 id="orgcd078d9"><span class="section-number-2">5</span> The Generalized Plant</h2>
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<div id="outline-container-orga8463c6" class="outline-2">
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<h2 id="orga8463c6"><span class="section-number-2">5</span> The Generalized Plant</h2>
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<div class="outline-text-2" id="text-5">
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<div id="org798bc0e" class="figure">
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<div id="org501fafe" class="figure">
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<p><img src="figs/general_plant.png" alt="general_plant.png" />
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</p>
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</div>
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<table id="org8df566a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org56ab58c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Notations for the general configuration</caption>
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<colgroup>
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@ -303,10 +303,10 @@ The maximum value of the magnitude over all frequencies does correspond to the \
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</div>
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</div>
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<div id="outline-container-org675f3d8" class="outline-2">
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<h2 id="org675f3d8"><span class="section-number-2">6</span> Problem Formulation</h2>
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<div id="outline-container-org8f2f474" class="outline-2">
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<h2 id="org8f2f474"><span class="section-number-2">6</span> Problem Formulation</h2>
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<div class="outline-text-2" id="text-6">
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<div class="important" id="orgaecd7ae">
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<div class="important" id="org3c999ad">
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<p>
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The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing controllers \(K\) which minimize
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</p>
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@ -317,7 +317,7 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
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</div>
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<div id="orgc5544e6" class="figure">
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<div id="orgbf7a5b3" class="figure">
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<p><img src="figs/general_control_names.png" alt="general_control_names.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>General Control Configuration</p>
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@ -326,17 +326,17 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
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</div>
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<div id="outline-container-org64eaa6d" class="outline-2">
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<h2 id="org64eaa6d"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2>
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<div id="outline-container-org6e6fa08" class="outline-2">
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<h2 id="org6e6fa08"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2>
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<div class="outline-text-2" id="text-7">
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<div id="orge2d651d" class="figure">
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<div id="orgbdf8949" class="figure">
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<p><img src="figs/classical_feedback.png" alt="classical_feedback.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Classical Feedback Architecture</p>
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</div>
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<table id="org101f1b2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org0716237" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 2:</span> Notations for the Classical Feedback Architecture</caption>
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<colgroup>
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@ -390,8 +390,8 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
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</div>
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</div>
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<div id="outline-container-orgee3fbb9" class="outline-2">
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<h2 id="orgee3fbb9"><span class="section-number-2">8</span> From a Classical Feedback Architecture to a Generalized Plant</h2>
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<div id="outline-container-org61c8d78" class="outline-2">
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<h2 id="org61c8d78"><span class="section-number-2">8</span> From a Classical Feedback Architecture to a Generalized Plant</h2>
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<div class="outline-text-2" id="text-8">
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<p>
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The procedure is:
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@ -401,13 +401,13 @@ The procedure is:
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<li>Remove \(K\) and rearrange the inputs and outputs</li>
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</ol>
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<div class="exampl" id="orgb5a13ef">
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<div class="exampl" id="orgf472923">
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<p>
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Let’s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orge163327">6</a>
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Let’s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgdcc8e73">6</a>
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</p>
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<div id="orge163327" class="figure">
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<div id="orgdcc8e73" class="figure">
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<p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Classical Feedback Control Architecture (Tracking)</p>
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@ -425,11 +425,11 @@ First, define the signals of the generalized plant:
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<p>
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Then, Remove \(K\) and rearrange the inputs and outputs.
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We obtain the generalized plant shown in Figure <a href="#orgfbdbe4a">7</a>.
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We obtain the generalized plant shown in Figure <a href="#org6782ec2">7</a>.
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</p>
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<div id="orgfbdbe4a" class="figure">
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<div id="org6782ec2" class="figure">
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<p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p>
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@ -449,12 +449,12 @@ Using Matlab, the generalized plant can be defined as follows:
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</div>
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</div>
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<div id="outline-container-orgc9be7b5" class="outline-2">
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<h2 id="orgc9be7b5"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2>
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<div id="outline-container-orgf0b775f" class="outline-2">
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<h2 id="orgf0b775f"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2>
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<div class="outline-text-2" id="text-9">
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</div>
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<div id="outline-container-orgf9aaff9" class="outline-3">
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<h3 id="orgf9aaff9"><span class="section-number-3">9.1</span> Introduction</h3>
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<div id="outline-container-org0691a02" class="outline-3">
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<h3 id="org0691a02"><span class="section-number-3">9.1</span> Introduction</h3>
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<div class="outline-text-3" id="text-9-1">
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<ul class="org-ul">
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<li><b>Reference tracking</b> Overshoot, Static error, Setling time
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@ -488,8 +488,8 @@ Using Matlab, the generalized plant can be defined as follows:
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</div>
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</div>
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<div id="outline-container-org42f3566" class="outline-2">
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<h2 id="org42f3566"><span class="section-number-2">10</span> Resources</h2>
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<div id="outline-container-org2d9c766" class="outline-2">
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<h2 id="org2d9c766"><span class="section-number-2">10</span> Resources</h2>
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<div class="outline-text-2" id="text-10">
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<p>
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<iframe width="1280" height="720" src="https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder="0" allowfullscreen></iframe>
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@ -503,7 +503,7 @@ Using Matlab, the generalized plant can be defined as follows:
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-11-25 mer. 19:37</p>
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<p class="date">Created: 2020-11-25 mer. 19:38</p>
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</div>
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</body>
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</html>
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16
index.org
16
index.org
@ -56,7 +56,7 @@ It consists of three steps:
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- _Specifications_: Response Time, Noise Rejection, Maximum input amplitude, Robustness, ...
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- _Mathematical Criteria_: Cost Function, Shape of TF
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# - Cost Function, Needed Bandwidth, Roll-off, ...
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# - $\Longrightarrow$ We will use the $\hinf$ Norm
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# - $\Longrightarrow$ We will use the $\mathcal{H}_\infty$ Norm
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3. *Synthesis*: research of $K$ that satisfies the specifications for the model of the system
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#+begin_src latex :file control-procedure.pdf
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@ -195,19 +195,19 @@ The maximum value of the magnitude over all frequencies does correspond to the $
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* $\mathcal{H}_\infty$ Synthesis
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*Optimization problem*:
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$\hinf$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the $\hinf$ norms of defined transfer functions are minimized.
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$\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the $\mathcal{H}_\infty$ norms of defined transfer functions are minimized.
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*Engineer work*:
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1. Write the problem as standard $\hinf$ problem
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2. Translate the specifications as $\hinf$ norms
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1. Write the problem as standard $\mathcal{H}_\infty$ problem
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2. Translate the specifications as $\mathcal{H}_\infty$ norms
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3. Make the synthesis and analyze the obtain controller
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4. Reduce the order of the controller for implementation
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*Many ways to use the $\hinf$ Synthesis*:
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- Traditional $\hinf$ Synthesis
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*Many ways to use the $\mathcal{H}_\infty$ Synthesis*:
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- Traditional $\mathcal{H}_\infty$ Synthesis
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- Mixed Sensitivity Loop Shaping
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- Fixed-Structure $\hinf$ Synthesis
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- Signal Based $\hinf$ Synthesis
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- Fixed-Structure $\mathcal{H}_\infty$ Synthesis
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- Signal Based $\mathcal{H}_\infty$ Synthesis
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* The Generalized Plant
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#+begin_src latex :file general_plant.pdf
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