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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. 19:34 -->
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<!-- 2020-11-25 mer. 19:37 -->
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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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<meta name="author" content="Dehaeze Thomas" />
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<script>MathJax = {
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tex: {
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tags: 'ams',
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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="#orgf9a4c6d">1. Introduction to the Control Methodology - Model Based Control</a></li>
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<li><a href="#org3fca6e1">2. Some Background: From Classical Control to Robust Control</a></li>
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<li><a href="#org2092347">3. The \(\mathcal{H}_\infty\) Norm</a></li>
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<li><a href="#org2331e77">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#org9b8faac">5. The Generalized Plant</a></li>
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<li><a href="#org8116ae5">6. Problem Formulation</a></li>
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<li><a href="#org42f9bcf">7. Classical feedback control and closed loop transfer functions</a></li>
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<li><a href="#orgf00feb8">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
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<li><a href="#org7c94ee9">9. Modern Interpretation of the Control Specifications</a>
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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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<ul>
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<li><a href="#orgfcb96b5">9.1. Introduction</a></li>
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<li><a href="#orgf9aaff9">9.1. Introduction</a></li>
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</ul>
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</li>
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<li><a href="#orgc9ce5a6">10. Resources</a></li>
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<li><a href="#org42f3566">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-orgf9a4c6d" class="outline-2">
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<h2 id="orgf9a4c6d"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2>
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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 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="#orgb1a2ae6">1</a>.
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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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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="orgb1a2ae6" class="figure">
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<div id="org3f1eee3" 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-org3fca6e1" class="outline-2">
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<h2 id="org3fca6e1"><span class="section-number-2">2</span> Some Background: From Classical Control to Robust Control</h2>
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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 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-org2092347" class="outline-2">
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<h2 id="org2092347"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2>
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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 class="outline-text-2" id="text-3">
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<div class="definition" id="org7770f0d">
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<div class="definition" id="org376138c">
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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="org5cdde3f">
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<div class="exampl" id="org4a6bc14">
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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="#orgd616903">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="#org109ee6e">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="orgd616903" class="figure">
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<div id="org109ee6e" 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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@ -216,8 +216,8 @@ 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-org2331e77" class="outline-2">
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<h2 id="org2331e77"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2>
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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 class="outline-text-2" id="text-4">
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<p>
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<b>Optimization problem</b>:
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@ -246,16 +246,16 @@ 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-org9b8faac" class="outline-2">
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<h2 id="org9b8faac"><span class="section-number-2">5</span> The Generalized Plant</h2>
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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 class="outline-text-2" id="text-5">
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<div id="orgd112ab1" class="figure">
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<div id="org798bc0e" 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="orgc823a19" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org8df566a" 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-org8116ae5" class="outline-2">
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<h2 id="org8116ae5"><span class="section-number-2">6</span> Problem Formulation</h2>
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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 class="outline-text-2" id="text-6">
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<div class="important" id="org179b25f">
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<div class="important" id="orgaecd7ae">
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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="orgcf0dd39" class="figure">
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<div id="orgc5544e6" 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-org42f9bcf" class="outline-2">
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<h2 id="org42f9bcf"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2>
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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 class="outline-text-2" id="text-7">
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<div id="orgb1f039f" class="figure">
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<div id="orge2d651d" 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="orgfaf4a42" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org101f1b2" 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-orgf00feb8" class="outline-2">
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<h2 id="orgf00feb8"><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-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 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="orgd38b593">
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<div class="exampl" id="orgb5a13ef">
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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="#orgaa07663">6</a>
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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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</p>
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<div id="orgaa07663" class="figure">
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<div id="orge163327" 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="#orgb73ca0b">7</a>.
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We obtain the generalized plant shown in Figure <a href="#orgfbdbe4a">7</a>.
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</p>
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<div id="orgb73ca0b" class="figure">
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<div id="orgfbdbe4a" 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-org7c94ee9" class="outline-2">
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<h2 id="org7c94ee9"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2>
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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 class="outline-text-2" id="text-9">
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</div>
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<div id="outline-container-orgfcb96b5" class="outline-3">
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<h3 id="orgfcb96b5"><span class="section-number-3">9.1</span> Introduction</h3>
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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 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-orgc9ce5a6" class="outline-2">
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<h2 id="orgc9ce5a6"><span class="section-number-2">10</span> Resources</h2>
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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 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. 19:34</p>
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<p class="date">Created: 2020-11-25 mer. 19:37</p>
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</div>
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</body>
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</html>
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@ -9,11 +9,8 @@
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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# #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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# #+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="/home/thomas/Cloud/thesis/matlab/research-home-page/css/style.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="/home/thomas/Cloud/thesis/matlab/research-home-page/js/script.js"></script>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+HTML_MATHJAX: align: center tagside: right font: TeX
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