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<title>Robust Control - \(\mathcal{H}_\infty\) Synthesis</title> <title>Robust Control - \(\mathcal{H}_\infty\) Synthesis</title>
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<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgf9a4c6d">1. Introduction to the Control Methodology - Model Based Control</a></li> <li><a href="#org4643b2e">1. Introduction to the Control Methodology - Model Based Control</a></li>
<li><a href="#org3fca6e1">2. Some Background: From Classical Control to Robust Control</a></li> <li><a href="#org8a683c3">2. Some Background: From Classical Control to Robust Control</a></li>
<li><a href="#org2092347">3. The \(\mathcal{H}_\infty\) Norm</a></li> <li><a href="#orgc683003">3. The \(\mathcal{H}_\infty\) Norm</a></li>
<li><a href="#org2331e77">4. \(\mathcal{H}_\infty\) Synthesis</a></li> <li><a href="#org23e7e4a">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#org9b8faac">5. The Generalized Plant</a></li> <li><a href="#orgcd078d9">5. The Generalized Plant</a></li>
<li><a href="#org8116ae5">6. Problem Formulation</a></li> <li><a href="#org675f3d8">6. Problem Formulation</a></li>
<li><a href="#org42f9bcf">7. Classical feedback control and closed loop transfer functions</a></li> <li><a href="#org64eaa6d">7. Classical feedback control and closed loop transfer functions</a></li>
<li><a href="#orgf00feb8">8. From a Classical Feedback Architecture to a Generalized Plant</a></li> <li><a href="#orgee3fbb9">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
<li><a href="#org7c94ee9">9. Modern Interpretation of the Control Specifications</a> <li><a href="#orgc9be7b5">9. Modern Interpretation of the Control Specifications</a>
<ul> <ul>
<li><a href="#orgfcb96b5">9.1. Introduction</a></li> <li><a href="#orgf9aaff9">9.1. Introduction</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgc9ce5a6">10. Resources</a></li> <li><a href="#org42f3566">10. Resources</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orgf9a4c6d" class="outline-2"> <div id="outline-container-org4643b2e" class="outline-2">
<h2 id="orgf9a4c6d"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2> <h2 id="org4643b2e"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#orgb1a2ae6">1</a>. The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org3f1eee3">1</a>.
It consists of three steps: It consists of three steps:
</p> </p>
<ol class="org-ol"> <ol class="org-ol">
@ -66,7 +66,7 @@ It consists of three steps:
</ol> </ol>
<div id="orgb1a2ae6" class="figure"> <div id="org3f1eee3" class="figure">
<p><img src="figs/control-procedure.png" alt="control-procedure.png" /> <p><img src="figs/control-procedure.png" alt="control-procedure.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Typical Methodoly for Model Based Control</p> <p><span class="figure-number">Figure 1: </span>Typical Methodoly for Model Based Control</p>
@ -78,8 +78,8 @@ In this document, we will mainly focus on steps 2 and 3.
</div> </div>
</div> </div>
<div id="outline-container-org3fca6e1" class="outline-2"> <div id="outline-container-org8a683c3" class="outline-2">
<h2 id="org3fca6e1"><span class="section-number-2">2</span> Some Background: From Classical Control to Robust Control</h2> <h2 id="org8a683c3"><span class="section-number-2">2</span> Some Background: From Classical Control to Robust Control</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
Classical Control (1930) Classical Control (1930)
@ -156,10 +156,10 @@ Robust Control (1980)
</div> </div>
</div> </div>
<div id="outline-container-org2092347" class="outline-2"> <div id="outline-container-orgc683003" class="outline-2">
<h2 id="org2092347"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2> <h2 id="orgc683003"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<div class="definition" id="org7770f0d"> <div class="definition" id="org376138c">
<p> <p>
The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
</p> </p>
@ -176,7 +176,7 @@ For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a
</div> </div>
<div class="exampl" id="org5cdde3f"> <div class="exampl" id="org4a6bc14">
<p> <p>
Let&rsquo;s define a plant dynamics: Let&rsquo;s define a plant dynamics:
</p> </p>
@ -201,12 +201,12 @@ And compute its \(\mathcal{H}_\infty\) norm using the <code>hinfnorm</code> func
<p> <p>
The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure <a href="#orgd616903">2</a>. The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure <a href="#org109ee6e">2</a>.
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. 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.
</p> </p>
<div id="orgd616903" class="figure"> <div id="org109ee6e" class="figure">
<p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" /> <p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p> <p><span class="figure-number">Figure 2: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p>
@ -216,8 +216,8 @@ The maximum value of the magnitude over all frequencies does correspond to the \
</div> </div>
</div> </div>
<div id="outline-container-org2331e77" class="outline-2"> <div id="outline-container-org23e7e4a" class="outline-2">
<h2 id="org2331e77"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2> <h2 id="org23e7e4a"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<b>Optimization problem</b>: <b>Optimization problem</b>:
@ -246,16 +246,16 @@ The maximum value of the magnitude over all frequencies does correspond to the \
</div> </div>
</div> </div>
<div id="outline-container-org9b8faac" class="outline-2"> <div id="outline-container-orgcd078d9" class="outline-2">
<h2 id="org9b8faac"><span class="section-number-2">5</span> The Generalized Plant</h2> <h2 id="orgcd078d9"><span class="section-number-2">5</span> The Generalized Plant</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<div id="orgd112ab1" class="figure"> <div id="org798bc0e" class="figure">
<p><img src="figs/general_plant.png" alt="general_plant.png" /> <p><img src="figs/general_plant.png" alt="general_plant.png" />
</p> </p>
</div> </div>
<table id="orgc823a19" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org8df566a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Notations for the general configuration</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Notations for the general configuration</caption>
<colgroup> <colgroup>
@ -303,10 +303,10 @@ The maximum value of the magnitude over all frequencies does correspond to the \
</div> </div>
</div> </div>
<div id="outline-container-org8116ae5" class="outline-2"> <div id="outline-container-org675f3d8" class="outline-2">
<h2 id="org8116ae5"><span class="section-number-2">6</span> Problem Formulation</h2> <h2 id="org675f3d8"><span class="section-number-2">6</span> Problem Formulation</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
<div class="important" id="org179b25f"> <div class="important" id="orgaecd7ae">
<p> <p>
The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing controllers \(K\) which minimize The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing controllers \(K\) which minimize
</p> </p>
@ -317,7 +317,7 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
</div> </div>
<div id="orgcf0dd39" class="figure"> <div id="orgc5544e6" class="figure">
<p><img src="figs/general_control_names.png" alt="general_control_names.png" /> <p><img src="figs/general_control_names.png" alt="general_control_names.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>General Control Configuration</p> <p><span class="figure-number">Figure 4: </span>General Control Configuration</p>
@ -326,17 +326,17 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
</div> </div>
<div id="outline-container-org42f9bcf" class="outline-2"> <div id="outline-container-org64eaa6d" class="outline-2">
<h2 id="org42f9bcf"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2> <h2 id="org64eaa6d"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2>
<div class="outline-text-2" id="text-7"> <div class="outline-text-2" id="text-7">
<div id="orgb1f039f" class="figure"> <div id="orge2d651d" class="figure">
<p><img src="figs/classical_feedback.png" alt="classical_feedback.png" /> <p><img src="figs/classical_feedback.png" alt="classical_feedback.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Classical Feedback Architecture</p> <p><span class="figure-number">Figure 5: </span>Classical Feedback Architecture</p>
</div> </div>
<table id="orgfaf4a42" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org101f1b2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Notations for the Classical Feedback Architecture</caption> <caption class="t-above"><span class="table-number">Table 2:</span> Notations for the Classical Feedback Architecture</caption>
<colgroup> <colgroup>
@ -390,8 +390,8 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
</div> </div>
</div> </div>
<div id="outline-container-orgf00feb8" class="outline-2"> <div id="outline-container-orgee3fbb9" class="outline-2">
<h2 id="orgf00feb8"><span class="section-number-2">8</span> From a Classical Feedback Architecture to a Generalized Plant</h2> <h2 id="orgee3fbb9"><span class="section-number-2">8</span> From a Classical Feedback Architecture to a Generalized Plant</h2>
<div class="outline-text-2" id="text-8"> <div class="outline-text-2" id="text-8">
<p> <p>
The procedure is: The procedure is:
@ -401,13 +401,13 @@ The procedure is:
<li>Remove \(K\) and rearrange the inputs and outputs</li> <li>Remove \(K\) and rearrange the inputs and outputs</li>
</ol> </ol>
<div class="exampl" id="orgd38b593"> <div class="exampl" id="orgb5a13ef">
<p> <p>
Let&rsquo;s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgaa07663">6</a> Let&rsquo;s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orge163327">6</a>
</p> </p>
<div id="orgaa07663" class="figure"> <div id="orge163327" class="figure">
<p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" /> <p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Classical Feedback Control Architecture (Tracking)</p> <p><span class="figure-number">Figure 6: </span>Classical Feedback Control Architecture (Tracking)</p>
@ -425,11 +425,11 @@ First, define the signals of the generalized plant:
<p> <p>
Then, Remove \(K\) and rearrange the inputs and outputs. Then, Remove \(K\) and rearrange the inputs and outputs.
We obtain the generalized plant shown in Figure <a href="#orgb73ca0b">7</a>. We obtain the generalized plant shown in Figure <a href="#orgfbdbe4a">7</a>.
</p> </p>
<div id="orgb73ca0b" class="figure"> <div id="orgfbdbe4a" class="figure">
<p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" /> <p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p> <p><span class="figure-number">Figure 7: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p>
@ -449,12 +449,12 @@ Using Matlab, the generalized plant can be defined as follows:
</div> </div>
</div> </div>
<div id="outline-container-org7c94ee9" class="outline-2"> <div id="outline-container-orgc9be7b5" class="outline-2">
<h2 id="org7c94ee9"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2> <h2 id="orgc9be7b5"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2>
<div class="outline-text-2" id="text-9"> <div class="outline-text-2" id="text-9">
</div> </div>
<div id="outline-container-orgfcb96b5" class="outline-3"> <div id="outline-container-orgf9aaff9" class="outline-3">
<h3 id="orgfcb96b5"><span class="section-number-3">9.1</span> Introduction</h3> <h3 id="orgf9aaff9"><span class="section-number-3">9.1</span> Introduction</h3>
<div class="outline-text-3" id="text-9-1"> <div class="outline-text-3" id="text-9-1">
<ul class="org-ul"> <ul class="org-ul">
<li><b>Reference tracking</b> Overshoot, Static error, Setling time <li><b>Reference tracking</b> Overshoot, Static error, Setling time
@ -488,8 +488,8 @@ Using Matlab, the generalized plant can be defined as follows:
</div> </div>
</div> </div>
<div id="outline-container-orgc9ce5a6" class="outline-2"> <div id="outline-container-org42f3566" class="outline-2">
<h2 id="orgc9ce5a6"><span class="section-number-2">10</span> Resources</h2> <h2 id="org42f3566"><span class="section-number-2">10</span> Resources</h2>
<div class="outline-text-2" id="text-10"> <div class="outline-text-2" id="text-10">
<p> <p>
<iframe width="1280" height="720" src="https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder="0" allowfullscreen></iframe> <iframe width="1280" height="720" src="https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder="0" allowfullscreen></iframe>
@ -503,7 +503,7 @@ Using Matlab, the generalized plant can be defined as follows:
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-25 mer. 19:34</p> <p class="date">Created: 2020-11-25 mer. 19:37</p>
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