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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-11-25 mer. 19:37 --> <!-- 2020-11-25 mer. 19:38 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Robust Control - \(\mathcal{H}_\infty\) Synthesis</title> <title>Robust Control - \(\mathcal{H}_\infty\) Synthesis</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -30,29 +30,29 @@
<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="#org4643b2e">1. Introduction to the Control Methodology - Model Based Control</a></li> <li><a href="#org983c7b8">1. Introduction to the Control Methodology - Model Based Control</a></li>
<li><a href="#org8a683c3">2. Some Background: From Classical Control to Robust Control</a></li> <li><a href="#org56e9b1e">2. Some Background: From Classical Control to Robust Control</a></li>
<li><a href="#orgc683003">3. The \(\mathcal{H}_\infty\) Norm</a></li> <li><a href="#org26e4f77">3. The \(\mathcal{H}_\infty\) Norm</a></li>
<li><a href="#org23e7e4a">4. \(\mathcal{H}_\infty\) Synthesis</a></li> <li><a href="#org7fece23">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#orgcd078d9">5. The Generalized Plant</a></li> <li><a href="#orga8463c6">5. The Generalized Plant</a></li>
<li><a href="#org675f3d8">6. Problem Formulation</a></li> <li><a href="#org8f2f474">6. Problem Formulation</a></li>
<li><a href="#org64eaa6d">7. Classical feedback control and closed loop transfer functions</a></li> <li><a href="#org6e6fa08">7. Classical feedback control and closed loop transfer functions</a></li>
<li><a href="#orgee3fbb9">8. From a Classical Feedback Architecture to a Generalized Plant</a></li> <li><a href="#org61c8d78">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
<li><a href="#orgc9be7b5">9. Modern Interpretation of the Control Specifications</a> <li><a href="#orgf0b775f">9. Modern Interpretation of the Control Specifications</a>
<ul> <ul>
<li><a href="#orgf9aaff9">9.1. Introduction</a></li> <li><a href="#org0691a02">9.1. Introduction</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org42f3566">10. Resources</a></li> <li><a href="#org2d9c766">10. Resources</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org4643b2e" class="outline-2"> <div id="outline-container-org983c7b8" class="outline-2">
<h2 id="org4643b2e"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2> <h2 id="org983c7b8"><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="#org3f1eee3">1</a>. The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org893c4a9">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="org3f1eee3" class="figure"> <div id="org893c4a9" 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-org8a683c3" class="outline-2"> <div id="outline-container-org56e9b1e" class="outline-2">
<h2 id="org8a683c3"><span class="section-number-2">2</span> Some Background: From Classical Control to Robust Control</h2> <h2 id="org56e9b1e"><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-orgc683003" class="outline-2"> <div id="outline-container-org26e4f77" class="outline-2">
<h2 id="orgc683003"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2> <h2 id="org26e4f77"><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="org376138c"> <div class="definition" id="org2a3b6b9">
<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="org4a6bc14"> <div class="exampl" id="org2014425">
<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="#org109ee6e">2</a>. The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure <a href="#org614e629">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="org109ee6e" class="figure"> <div id="org614e629" 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>
@ -216,46 +216,46 @@ The maximum value of the magnitude over all frequencies does correspond to the \
</div> </div>
</div> </div>
<div id="outline-container-org23e7e4a" class="outline-2"> <div id="outline-container-org7fece23" class="outline-2">
<h2 id="org23e7e4a"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2> <h2 id="org7fece23"><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>:
\(\hinf\) 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. \(\mathcal{H}_\infty\) 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.
</p> </p>
<p> <p>
<b>Engineer work</b>: <b>Engineer work</b>:
</p> </p>
<ol class="org-ol"> <ol class="org-ol">
<li>Write the problem as standard \(\hinf\) problem</li> <li>Write the problem as standard \(\mathcal{H}_\infty\) problem</li>
<li>Translate the specifications as \(\hinf\) norms</li> <li>Translate the specifications as \(\mathcal{H}_\infty\) norms</li>
<li>Make the synthesis and analyze the obtain controller</li> <li>Make the synthesis and analyze the obtain controller</li>
<li>Reduce the order of the controller for implementation</li> <li>Reduce the order of the controller for implementation</li>
</ol> </ol>
<p> <p>
<b>Many ways to use the \(\hinf\) Synthesis</b>: <b>Many ways to use the \(\mathcal{H}_\infty\) Synthesis</b>:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Traditional \(\hinf\) Synthesis</li> <li>Traditional \(\mathcal{H}_\infty\) Synthesis</li>
<li>Mixed Sensitivity Loop Shaping</li> <li>Mixed Sensitivity Loop Shaping</li>
<li>Fixed-Structure \(\hinf\) Synthesis</li> <li>Fixed-Structure \(\mathcal{H}_\infty\) Synthesis</li>
<li>Signal Based \(\hinf\) Synthesis</li> <li>Signal Based \(\mathcal{H}_\infty\) Synthesis</li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orgcd078d9" class="outline-2"> <div id="outline-container-orga8463c6" class="outline-2">
<h2 id="orgcd078d9"><span class="section-number-2">5</span> The Generalized Plant</h2> <h2 id="orga8463c6"><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="org798bc0e" class="figure"> <div id="org501fafe" 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="org8df566a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org56ab58c" 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-org675f3d8" class="outline-2"> <div id="outline-container-org8f2f474" class="outline-2">
<h2 id="org675f3d8"><span class="section-number-2">6</span> Problem Formulation</h2> <h2 id="org8f2f474"><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="orgaecd7ae"> <div class="important" id="org3c999ad">
<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="orgc5544e6" class="figure"> <div id="orgbf7a5b3" 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-org64eaa6d" class="outline-2"> <div id="outline-container-org6e6fa08" class="outline-2">
<h2 id="org64eaa6d"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2> <h2 id="org6e6fa08"><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="orge2d651d" class="figure"> <div id="orgbdf8949" 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="org101f1b2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org0716237" 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-orgee3fbb9" class="outline-2"> <div id="outline-container-org61c8d78" class="outline-2">
<h2 id="orgee3fbb9"><span class="section-number-2">8</span> From a Classical Feedback Architecture to a Generalized Plant</h2> <h2 id="org61c8d78"><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="orgb5a13ef"> <div class="exampl" id="orgf472923">
<p> <p>
Let&rsquo;s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orge163327">6</a> Let&rsquo;s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgdcc8e73">6</a>
</p> </p>
<div id="orge163327" class="figure"> <div id="orgdcc8e73" 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="#orgfbdbe4a">7</a>. We obtain the generalized plant shown in Figure <a href="#org6782ec2">7</a>.
</p> </p>
<div id="orgfbdbe4a" class="figure"> <div id="org6782ec2" 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-orgc9be7b5" class="outline-2"> <div id="outline-container-orgf0b775f" class="outline-2">
<h2 id="orgc9be7b5"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2> <h2 id="orgf0b775f"><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-orgf9aaff9" class="outline-3"> <div id="outline-container-org0691a02" class="outline-3">
<h3 id="orgf9aaff9"><span class="section-number-3">9.1</span> Introduction</h3> <h3 id="org0691a02"><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-org42f3566" class="outline-2"> <div id="outline-container-org2d9c766" class="outline-2">
<h2 id="org42f3566"><span class="section-number-2">10</span> Resources</h2> <h2 id="org2d9c766"><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. 19:37</p> <p class="date">Created: 2020-11-25 mer. 19:38</p>
</div> </div>
</body> </body>
</html> </html>

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@ -56,7 +56,7 @@ It consists of three steps:
- _Specifications_: Response Time, Noise Rejection, Maximum input amplitude, Robustness, ... - _Specifications_: Response Time, Noise Rejection, Maximum input amplitude, Robustness, ...
- _Mathematical Criteria_: Cost Function, Shape of TF - _Mathematical Criteria_: Cost Function, Shape of TF
# - Cost Function, Needed Bandwidth, Roll-off, ... # - Cost Function, Needed Bandwidth, Roll-off, ...
# - $\Longrightarrow$ We will use the $\hinf$ Norm # - $\Longrightarrow$ We will use the $\mathcal{H}_\infty$ Norm
3. *Synthesis*: research of $K$ that satisfies the specifications for the model of the system 3. *Synthesis*: research of $K$ that satisfies the specifications for the model of the system
#+begin_src latex :file control-procedure.pdf #+begin_src latex :file control-procedure.pdf
@ -195,19 +195,19 @@ The maximum value of the magnitude over all frequencies does correspond to the $
* $\mathcal{H}_\infty$ Synthesis * $\mathcal{H}_\infty$ Synthesis
*Optimization problem*: *Optimization problem*:
$\hinf$ 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. $\mathcal{H}_\infty$ 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.
*Engineer work*: *Engineer work*:
1. Write the problem as standard $\hinf$ problem 1. Write the problem as standard $\mathcal{H}_\infty$ problem
2. Translate the specifications as $\hinf$ norms 2. Translate the specifications as $\mathcal{H}_\infty$ norms
3. Make the synthesis and analyze the obtain controller 3. Make the synthesis and analyze the obtain controller
4. Reduce the order of the controller for implementation 4. Reduce the order of the controller for implementation
*Many ways to use the $\hinf$ Synthesis*: *Many ways to use the $\mathcal{H}_\infty$ Synthesis*:
- Traditional $\hinf$ Synthesis - Traditional $\mathcal{H}_\infty$ Synthesis
- Mixed Sensitivity Loop Shaping - Mixed Sensitivity Loop Shaping
- Fixed-Structure $\hinf$ Synthesis - Fixed-Structure $\mathcal{H}_\infty$ Synthesis
- Signal Based $\hinf$ Synthesis - Signal Based $\mathcal{H}_\infty$ Synthesis
* The Generalized Plant * The Generalized Plant
#+begin_src latex :file general_plant.pdf #+begin_src latex :file general_plant.pdf