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Add example system

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Thomas Dehaeze 2020-11-27 18:26:06 +01:00
parent 178b273105
commit 1b8b824d9f
15 changed files with 1307 additions and 63 deletions
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<head>
<!-- 2020-11-25 mer. 19:38 -->
<!-- 2020-11-27 ven. 18:20 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Robust Control - \(\mathcal{H}_\infty\) Synthesis</title>
<meta name="generator" content="Org mode" />
@ -30,29 +30,46 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org983c7b8">1. Introduction to the Control Methodology - Model Based Control</a></li>
<li><a href="#org56e9b1e">2. Some Background: From Classical Control to Robust Control</a></li>
<li><a href="#org26e4f77">3. The \(\mathcal{H}_\infty\) Norm</a></li>
<li><a href="#org7fece23">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#orga8463c6">5. The Generalized Plant</a></li>
<li><a href="#org8f2f474">6. Problem Formulation</a></li>
<li><a href="#org6e6fa08">7. Classical feedback control and closed loop transfer functions</a></li>
<li><a href="#org61c8d78">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
<li><a href="#orgf0b775f">9. Modern Interpretation of the Control Specifications</a>
<li><a href="#org482cee2">1. Introduction to the Control Methodology - Model Based Control</a>
<ul>
<li><a href="#org0691a02">9.1. Introduction</a></li>
<li><a href="#org279a16f">1.1. Control Methodology</a></li>
<li><a href="#orgca6e3d9">1.2. Some Background: From Classical Control to Robust Control</a></li>
<li><a href="#orgc1a54dd">1.3. Example System</a></li>
</ul>
</li>
<li><a href="#org2d9c766">10. Resources</a></li>
<li><a href="#org1336132">2. Classical Open Loop Shaping</a>
<ul>
<li><a href="#org1c9ddc9">2.1. Introduction ot Open Loop Shaping</a></li>
<li><a href="#org3cd2ec2">2.2. Example of Open Loop Shaping</a></li>
<li><a href="#orgafc190d">2.3. \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
<li><a href="#org386c720">2.4. Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
</ul>
</li>
<li><a href="#orgcf6b2a6">3. The \(\mathcal{H}_\infty\) Norm</a></li>
<li><a href="#org5bc80a8">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#org3a241a5">5. The Generalized Plant</a></li>
<li><a href="#org49fe7df">6. Problem Formulation</a></li>
<li><a href="#orgf671631">7. Classical feedback control and closed loop transfer functions</a></li>
<li><a href="#orgd118710">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
<li><a href="#orgdd1c5fa">9. Modern Interpretation of the Control Specifications</a>
<ul>
<li><a href="#org00cad5d">9.1. Introduction</a></li>
</ul>
</li>
<li><a href="#org0e07a10">10. Resources</a></li>
</ul>
</div>
</div>
<div id="outline-container-org983c7b8" class="outline-2">
<h2 id="org983c7b8"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2>
<div id="outline-container-org482cee2" class="outline-2">
<h2 id="org482cee2"><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>
<div id="outline-container-org279a16f" class="outline-3">
<h3 id="org279a16f"><span class="section-number-3">1.1</span> Control Methodology</h3>
<div class="outline-text-3" id="text-1-1">
<p>
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org893c4a9">1</a>.
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#org3328399">1</a>.
It consists of three steps:
</p>
<ol class="org-ol">
@ -66,7 +83,7 @@ It consists of three steps:
</ol>
<div id="org893c4a9" class="figure">
<div id="org3328399" class="figure">
<p><img src="figs/control-procedure.png" alt="control-procedure.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Typical Methodoly for Model Based Control</p>
@ -78,9 +95,9 @@ In this document, we will mainly focus on steps 2 and 3.
</div>
</div>
<div id="outline-container-org56e9b1e" class="outline-2">
<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 id="outline-container-orgca6e3d9" class="outline-3">
<h3 id="orgca6e3d9"><span class="section-number-3">1.2</span> Some Background: From Classical Control to Robust Control</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Classical Control (1930)
</p>
@ -156,10 +173,363 @@ Robust Control (1980)
</div>
</div>
<div id="outline-container-org26e4f77" class="outline-2">
<h2 id="org26e4f77"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2>
<div id="outline-container-orgc1a54dd" class="outline-3">
<h3 id="orgc1a54dd"><span class="section-number-3">1.3</span> Example System</h3>
<div class="outline-text-3" id="text-1-3">
<p>
Let&rsquo;s consider the test-system shown in Figure <a href="#orgbec3f57">2</a>.
The notations used are listed in Table <a href="#orgf10115b">1</a>.
</p>
<div id="orgbec3f57" class="figure">
<p><img src="figs/mech_sys_1dof_inertial_contr.png" alt="mech_sys_1dof_inertial_contr.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Test System consisting of a payload with a mass \(m\) on top of an active system with a stiffness \(k\), damping \(c\) and an actuator.</p>
</div>
<table id="orgf10115b" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Example system variables</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"><b>Notation</b></th>
<th scope="col" class="org-left"><b>Description</b></th>
<th scope="col" class="org-left"><b>Value</b></th>
<th scope="col" class="org-left"><b>Unit</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(m\)</td>
<td class="org-left">Payload&rsquo;s mass to position / isolate</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[kg]</td>
</tr>
<tr>
<td class="org-left">\(k\)</td>
<td class="org-left">Stiffness of the suspension system</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[N/m]</td>
</tr>
<tr>
<td class="org-left">\(c\)</td>
<td class="org-left">Damping coefficient of the suspension system</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[N/(m/s)]</td>
</tr>
<tr>
<td class="org-left">\(y\)</td>
<td class="org-left">Payload absolute displacement (measured by an inertial sensor)</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[m]</td>
</tr>
<tr>
<td class="org-left">\(d\)</td>
<td class="org-left">Ground displacement, it acts as a disturbance</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[m]</td>
</tr>
<tr>
<td class="org-left">\(u\)</td>
<td class="org-left">Actuator force</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[N]</td>
</tr>
<tr>
<td class="org-left">\(r\)</td>
<td class="org-left">Wanted position of the mass (the reference)</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[m]</td>
</tr>
<tr>
<td class="org-left">\(\epsilon = r - y\)</td>
<td class="org-left">Position error</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">[m]</td>
</tr>
<tr>
<td class="org-left">\(K\)</td>
<td class="org-left">Feedback controller</td>
<td class="org-left">to be designed</td>
<td class="org-left">[N/m]</td>
</tr>
</tbody>
</table>
<div class="exercice" id="org09b2f15">
<p>
Derive the following open-loop transfer functions:
</p>
\begin{align}
G(s) &= \frac{y}{u} \\
G_d(s) &= \frac{y}{d}
\end{align}
<p>
<b>Hint:</b> You can follow this generic procedure:
</p>
<ol class="org-ol">
<li>List all applied forces ot the mass: Actuator force, Stiffness force (Hooke&rsquo;s law), &#x2026;</li>
<li>Apply the Newton&rsquo;s Second Law on the payload
\[ m \ddot{y} = \Sigma F \]</li>
<li>Transform the differential equations into the Laplace domain:
\[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]</li>
<li>Write \(y(s)\) as a function of \(u(s)\) and \(w(s)\)</li>
</ol>
</div>
<p>
Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure <a href="#orgbec3f57">2</a> into a classical feedback form as shown in Figure <a href="#orge09e85e">4</a>.
</p>
<div id="orge1cf983" class="figure">
<p><img src="figs/classical_feedback_test_system.png" alt="classical_feedback_test_system.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Block diagram corresponding to the example system</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org1336132" class="outline-2">
<h2 id="org1336132"><span class="section-number-2">2</span> Classical Open Loop Shaping</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org1c9ddc9" class="outline-3">
<h3 id="org1c9ddc9"><span class="section-number-3">2.1</span> Introduction ot Open Loop Shaping</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Usually, the controller \(K(s)\) is designed such that the loop gain \(L(s)\) has desirable shape.
This technique is called <b>Open Loop Shaping</b>.
</p>
<div class="inlinetask">
<b>Explain why the Loop gain si an important &ldquo;value&rdquo;</b><br />
<p>
For instance example all the specifications can usually be explained in terms of the open loop gain.
</p>
</div>
<div id="orge09e85e" class="figure">
<p><img src="figs/open_loop_shaping.png" alt="open_loop_shaping.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Classical Feedback Architecture</p>
</div>
<p>
This is usually done manually has the loop gain \(L(s)\) depends linearly of \(K(s)\):
</p>
\begin{equation}
L(s) = G(s) K(s)
\end{equation}
<ul class="org-ul">
<li>where \(L(s)\) is called the <b>Loop Gain Transfer Function</b></li>
</ul>
<p>
\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that its product with \(G(s)\) has wanted shape.
</p>
</div>
</div>
<div id="outline-container-org3cd2ec2" class="outline-3">
<h3 id="org3cd2ec2"><span class="section-number-3">2.2</span> Example of Open Loop Shaping</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-matlab">k = 1e<span class="org-type">-</span>6;
m = 10;
c = 10;
G =
</pre>
</div>
<div id="org846352b" class="figure">
<p><img src="figs/bode_plot_example_afm.png" alt="bode_plot_example_afm.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Bode plot of the plant \(G(s)\)</p>
</div>
<p>
Specifications:
</p>
<ul class="org-ul">
<li><b>Performance</b>: Bandwidth of approximately 50Hz</li>
<li><b>Noise Attenuation</b>: Roll-off of -40dB/decade past 250Hz</li>
<li><b>Robustness</b>: Gain margin &gt; 5dB and Phase margin &gt; 40 deg</li>
</ul>
<div class="exercice" id="orge257cef">
<p>
Using <code>SISOTOOL</code>, design a controller that fulfill the specifications.
</p>
<div class="org-src-container">
<pre class="src src-matlab">sisotool(G)
</pre>
</div>
</div>
<p>
In order to have the wanted Roll-off, two integrators are used, a lead is also added to have sufficient phase margin.
</p>
<p>
The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure <a href="#orgd8a3cda">6</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = 6e4 <span class="org-type">*</span> ...<span class="org-comment"> % Gain</span>
1<span class="org-type">/</span>(s<span class="org-type">^</span>2) <span class="org-type">*</span> ...<span class="org-comment"> % Double Integrator</span>
(1 <span class="org-type">+</span> s<span class="org-type">/</span>111)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>888); <span class="org-comment">% Lead</span>
</pre>
</div>
<div id="orgd8a3cda" class="figure">
<p><img src="figs/loop_gain_manual_afm.png" alt="loop_gain_manual_afm.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Bode Plot of the obtained Loop Gain \(L(s) = G(s) K(s)\)</p>
</div>
<p>
And we can verify that we have the wanted stability margins:
</p>
<div class="org-src-container">
<pre class="src src-matlab">[Gm, Pm, <span class="org-type">~</span>, Wc] = margin(G<span class="org-type">*</span>K)
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">Value</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Gain Margin [dB]</td>
<td class="org-right">7.2</td>
</tr>
<tr>
<td class="org-left">Phase Margin [deg]</td>
<td class="org-right">48.1</td>
</tr>
<tr>
<td class="org-left">Crossover [Hz]</td>
<td class="org-right">50.7</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgafc190d" class="outline-3">
<h3 id="orgafc190d"><span class="section-number-3">2.3</span> \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
<div class="outline-text-3" id="text-2-3">
<p>
The Open Loop Shaping synthesis can be performed using the \(\mathcal{H}_\infty\) Synthesis.
</p>
<p>
Even though we will not go into details, we will provide one example.
</p>
<p>
Using Matlab, the \(\mathcal{H}_\infty\) synthesis of a controller based on the wanted open loop shape can be performed using the <code>loopsyn</code> command:
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = loopsyn(G, Gd);
</pre>
</div>
<p>
where:
</p>
<ul class="org-ul">
<li><code>G</code> is the (LTI) plant</li>
<li><code>Gd</code> is the wanted loop shape</li>
<li><code>K</code> is the synthesize controller</li>
</ul>
<div class="seealso" id="org3c008e3">
<p>
Matlab documentation of <code>loopsyn</code> (<a href="https://www.mathworks.com/help/robust/ref/loopsyn.html">link</a>).
</p>
</div>
</div>
</div>
<div id="outline-container-org386c720" class="outline-3">
<h3 id="org386c720"><span class="section-number-3">2.4</span> Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
<div class="outline-text-3" id="text-2-4">
<p>
Let&rsquo;s re-use the previous plant.
</p>
<p>
Translate the specification into the wanted shape of the open loop gain.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G = tf(16,[1 0.16 16]);
Gd = 3.7e4<span class="org-type">*</span>1<span class="org-type">/</span>s<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>20)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>220)<span class="org-type">*</span>1<span class="org-type">/</span>(s <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>500);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">[K,CL,GAM,INFO] = loopsyn(G, Gd);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">bodeFig({K})
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgcf6b2a6" class="outline-2">
<h2 id="orgcf6b2a6"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2>
<div class="outline-text-2" id="text-3">
<div class="definition" id="org2a3b6b9">
<div class="definition" id="org86c267f">
<p>
The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
</p>
@ -176,7 +546,7 @@ For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a
</div>
<div class="exampl" id="org2014425">
<div class="exampl" id="org9ae7fd2">
<p>
Let&rsquo;s define a plant dynamics:
</p>
@ -201,23 +571,23 @@ And compute its \(\mathcal{H}_\infty\) norm using the <code>hinfnorm</code> func
<p>
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 magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure <a href="#org220c414">7</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.
</p>
<div id="org614e629" class="figure">
<div id="org220c414" class="figure">
<p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" />
</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 7: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org7fece23" class="outline-2">
<h2 id="org7fece23"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2>
<div id="outline-container-org5bc80a8" class="outline-2">
<h2 id="org5bc80a8"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2>
<div class="outline-text-2" id="text-4">
<p>
<b>Optimization problem</b>:
@ -246,17 +616,17 @@ The maximum value of the magnitude over all frequencies does correspond to the \
</div>
</div>
<div id="outline-container-orga8463c6" class="outline-2">
<h2 id="orga8463c6"><span class="section-number-2">5</span> The Generalized Plant</h2>
<div id="outline-container-org3a241a5" class="outline-2">
<h2 id="org3a241a5"><span class="section-number-2">5</span> The Generalized Plant</h2>
<div class="outline-text-2" id="text-5">
<div id="org501fafe" class="figure">
<div id="orgf05141d" class="figure">
<p><img src="figs/general_plant.png" alt="general_plant.png" />
</p>
</div>
<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>
<table id="orgfb53780" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Notations for the general configuration</caption>
<colgroup>
<col class="org-left" />
@ -303,10 +673,10 @@ The maximum value of the magnitude over all frequencies does correspond to the \
</div>
</div>
<div id="outline-container-org8f2f474" class="outline-2">
<h2 id="org8f2f474"><span class="section-number-2">6</span> Problem Formulation</h2>
<div id="outline-container-org49fe7df" class="outline-2">
<h2 id="org49fe7df"><span class="section-number-2">6</span> Problem Formulation</h2>
<div class="outline-text-2" id="text-6">
<div class="important" id="org3c999ad">
<div class="important" id="orgcec66b9">
<p>
The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing controllers \(K\) which minimize
</p>
@ -317,27 +687,27 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
</div>
<div id="orgbf7a5b3" class="figure">
<div id="org1e975ee" class="figure">
<p><img src="figs/general_control_names.png" alt="general_control_names.png" />
</p>
<p><span class="figure-number">Figure 4: </span>General Control Configuration</p>
<p><span class="figure-number">Figure 9: </span>General Control Configuration</p>
</div>
</div>
</div>
<div id="outline-container-org6e6fa08" class="outline-2">
<h2 id="org6e6fa08"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2>
<div id="outline-container-orgf671631" class="outline-2">
<h2 id="orgf671631"><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 id="orgbdf8949" class="figure">
<div id="orgd59dc12" class="figure">
<p><img src="figs/classical_feedback.png" alt="classical_feedback.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Classical Feedback Architecture</p>
<p><span class="figure-number">Figure 10: </span>Classical Feedback Architecture</p>
</div>
<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>
<table id="org111f2c5" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Notations for the Classical Feedback Architecture</caption>
<colgroup>
<col class="org-left" />
@ -390,8 +760,8 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
</div>
</div>
<div id="outline-container-org61c8d78" class="outline-2">
<h2 id="org61c8d78"><span class="section-number-2">8</span> From a Classical Feedback Architecture to a Generalized Plant</h2>
<div id="outline-container-orgd118710" class="outline-2">
<h2 id="orgd118710"><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">
<p>
The procedure is:
@ -401,16 +771,16 @@ The procedure is:
<li>Remove \(K\) and rearrange the inputs and outputs</li>
</ol>
<div class="exampl" id="orgf472923">
<div class="exampl" id="org211c0bc">
<p>
Let&rsquo;s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgdcc8e73">6</a>
Let&rsquo;s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgbec59d7">11</a>
</p>
<div id="orgdcc8e73" class="figure">
<div id="orgbec59d7" class="figure">
<p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Classical Feedback Control Architecture (Tracking)</p>
<p><span class="figure-number">Figure 11: </span>Classical Feedback Control Architecture (Tracking)</p>
</div>
<p>
@ -425,14 +795,14 @@ First, define the signals of the generalized plant:
<p>
Then, Remove \(K\) and rearrange the inputs and outputs.
We obtain the generalized plant shown in Figure <a href="#org6782ec2">7</a>.
We obtain the generalized plant shown in Figure <a href="#org64eccd4">12</a>.
</p>
<div id="org6782ec2" class="figure">
<div id="org64eccd4" class="figure">
<p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" />
</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 12: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p>
</div>
<p>
@ -449,12 +819,12 @@ Using Matlab, the generalized plant can be defined as follows:
</div>
</div>
<div id="outline-container-orgf0b775f" class="outline-2">
<h2 id="orgf0b775f"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2>
<div id="outline-container-orgdd1c5fa" class="outline-2">
<h2 id="orgdd1c5fa"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2>
<div class="outline-text-2" id="text-9">
</div>
<div id="outline-container-org0691a02" class="outline-3">
<h3 id="org0691a02"><span class="section-number-3">9.1</span> Introduction</h3>
<div id="outline-container-org00cad5d" class="outline-3">
<h3 id="org00cad5d"><span class="section-number-3">9.1</span> Introduction</h3>
<div class="outline-text-3" id="text-9-1">
<ul class="org-ul">
<li><b>Reference tracking</b> Overshoot, Static error, Setling time
@ -488,22 +858,22 @@ Using Matlab, the generalized plant can be defined as follows:
</div>
</div>
<div id="outline-container-org2d9c766" class="outline-2">
<h2 id="org2d9c766"><span class="section-number-2">10</span> Resources</h2>
<div id="outline-container-org0e07a10" class="outline-2">
<h2 id="org0e07a10"><span class="section-number-2">10</span> Resources</h2>
<div class="outline-text-2" id="text-10">
<p>
<iframe width="1280" height="720" src="https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder="0" allowfullscreen></iframe>
<div class="yt"><iframe width="100%" height="100%" src="https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder="0" allowfullscreen></iframe></div>
</p>
<p>
<iframe width="1280" height="720" src="https://www.youtube.com/embed/?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o" frameborder="0" allowfullscreen></iframe>
<div class="yt"><iframe width="100%" height="100%" src="https://www.youtube.com/embed/?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o" frameborder="0" allowfullscreen></iframe></div>
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-25 mer. 19:38</p>
<p class="date">Created: 2020-11-27 ven. 18:20</p>
</div>
</body>
</html>

353
index.org
View File

@ -38,6 +38,7 @@
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* Introduction :ignore:
* Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
@ -48,6 +49,7 @@
#+end_src
* Introduction to the Control Methodology - Model Based Control
** Control Methodology
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]].
It consists of three steps:
@ -91,7 +93,7 @@ It consists of three steps:
In this document, we will mainly focus on steps 2 and 3.
* Some Background: From Classical Control to Robust Control
** Some Background: From Classical Control to Robust Control
Classical Control (1930)
- Tools:
@ -136,6 +138,355 @@ Robust Control (1980)
- Requires the knowledge of specific tools
- Need a reasonably good model of the system
** Example System
Let's consider the test-system shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
The notations used are listed in Table [[tab:example_notations]].
#+begin_src latex :file mech_sys_1dof_inertial_contr.pdf
\begin{tikzpicture}
% Parameters
\def\massw{3}
\def\massh{1}
\def\spaceh{1.8}
% Ground
\draw[] (-0.5*\massw, 0) -- (0.5*\massw, 0);
% Mass
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5](m){$m$};
% Spring, Damper, and Actuator
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] ( 0, 0) -- ( 0, \spaceh) node[midway, left=0.3]{$c$};
\draw[actuator] ( 0.3*\massw, 0) -- (0.3*\massw, \spaceh) node[midway](F){};
% Displacements
\draw[dashed] (0.5*\massw, 0) -- ++(0.5, 0);
\draw[->] (0.6*\massw, 0) -- ++(0, 0.5) node[below right]{$d$};
% Inertial Sensor
\node[inertialsensor] (inertials) at (0.5*\massw, \spaceh+\massh){};
\node[addb={+}{-}{}{}{}, right=0.8 of inertials] (subf) {};
\node[block, below=0.4 of subf] (K){$K(s)$};
\draw[->] (inertials.east) node[above right]{$y$} -- (subf.west);
\draw[->] (subf.south) -- (K.north) node[above right]{$\epsilon$};
\draw[<-] (subf.north) -- ++(0, 0.6) node[below right]{$r$};
\draw[->] (K.south) |- (F.east) node[above right]{$u$};
\end{tikzpicture}
#+end_src
#+name: fig:mech_sys_1dof_inertial_contr
#+caption: Test System consisting of a payload with a mass $m$ on top of an active system with a stiffness $k$, damping $c$ and an actuator. A feedback controller $K(s)$ is added to position / isolate the payload.
#+RESULTS:
[[file:figs/mech_sys_1dof_inertial_contr.png]]
#+name: tab:example_notations
#+caption: Example system variables
| *Notation* | *Description* | *Value* | *Unit* |
|--------------------+----------------------------------------------------------------+----------------+-----------|
| $m$ | Payload's mass to position / isolate | | [kg] |
| $k$ | Stiffness of the suspension system | | [N/m] |
| $c$ | Damping coefficient of the suspension system | | [N/(m/s)] |
| $y$ | Payload absolute displacement (measured by an inertial sensor) | | [m] |
| $d$ | Ground displacement, it acts as a disturbance | | [m] |
| $u$ | Actuator force | | [N] |
| $r$ | Wanted position of the mass (the reference) | | [m] |
| $\epsilon = r - y$ | Position error | | [m] |
| $K$ | Feedback controller | to be designed | [N/m] |
#+begin_exercice
Derive the following open-loop transfer functions:
\begin{align}
G(s) &= \frac{y}{u} \\
G_d(s) &= \frac{y}{d}
\end{align}
*Hint:* You can follow this generic procedure:
1. List all applied forces ot the mass: Actuator force, Stiffness force (Hooke's law), ...
2. Apply the Newton's Second Law on the payload
\[ m \ddot{y} = \Sigma F \]
3. Transform the differential equations into the Laplace domain:
\[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]
4. Write $y(s)$ as a function of $u(s)$ and $w(s)$
#+end_exercice
Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure [[fig:mech_sys_1dof_inertial_contr]] into a classical feedback form as shown in Figure [[fig:open_loop_shaping]].
#+begin_src latex :file classical_feedback_test_system.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=0.8 of addfb] (K){$K(s)$};
\node[block, right=0.8 of K] (G){$G(s)$};
\node[addb={+}{}{}{}{}, right=0.8 of G] (addd){};
\node[block, above=0.5 of addd] (Gd){$G_d(s)$};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- (addd.west);
\draw[<-] (Gd.north) -- ++(0, 0.8) node[below right]{$d$};
\draw[->] (Gd.south) -- (addd.north);
\draw[->] (addd.east) -- ++(1.2, 0);
\draw[->] ($(addd.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -1.0) -| (addfb.south);
\end{tikzpicture}
#+end_src
#+name: fig:classical_feedback_test_system
#+caption: Block diagram corresponding to the example system
#+RESULTS:
[[file:figs/classical_feedback_test_system.png]]
#+begin_src matlab
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 16; % Mass [kg]
#+end_src
#+begin_src matlab
G = 1/(m*s^2 + c*s + k);
Gd = (c*s + k)/(m*s^2 + c*s + k);
#+end_src
* Classical Open Loop Shaping
** Introduction ot Open Loop Shaping
Usually, the controller $K(s)$ is designed such that the loop gain $L(s)$ has desirable shape.
This technique is called *Open Loop Shaping*.
*************** TODO Explain why the Loop gain si an important "value"
For instance example all the specifications can usually be explained in terms of the open loop gain.
*************** END
#+begin_src latex :file open_loop_shaping.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
\node[block, right=0.8 of addsub] (K) {$K(s)$};
\node[below] at (K.south) {Controller};
\node[block, right=0.8 of K] (G) {$G(s)$};
\node[below] at (G.south) {Plant};
\draw[<-] (addsub.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addsub) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(0.8, 0) node[above left]{$y$};
\draw[] ($(G.east) + (0.5, 0)$) -- ++(0, -1.4);
\draw[->] ($(G.east) + (0.5, -1.4)$) -| (addsub.south);
\draw [decoration={brace, raise=5pt}, decorate] (K.north west) -- node[above=6pt]{$L(s)$} (G.north east);
\end{tikzpicture}
#+end_src
#+name: fig:open_loop_shaping
#+caption: Classical Feedback Architecture
#+RESULTS:
[[file:figs/open_loop_shaping.png]]
This is usually done manually has the loop gain $L(s)$ depends linearly of $K(s)$:
\begin{equation}
L(s) = G(s) K(s)
\end{equation}
- where $L(s)$ is called the *Loop Gain Transfer Function*
$K(s)$ then consists of a combination of leads, lags, notches, etc. such that its product with $G(s)$ has wanted shape.
** Example of Open Loop Shaping
#+begin_src matlab
k = 1e-6;
m = 10;
c = 10;
G =
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-270, 90]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_example_afm.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:bode_plot_example_afm
#+caption: Bode plot of the plant $G(s)$
#+RESULTS:
[[file:figs/bode_plot_example_afm.png]]
Specifications:
- *Performance*: Bandwidth of approximately 50Hz
- *Noise Attenuation*: Roll-off of -40dB/decade past 250Hz
- *Robustness*: Gain margin > 5dB and Phase margin > 40 deg
#+begin_exercice
Using =SISOTOOL=, design a controller that fulfill the specifications.
#+begin_src matlab :eval no
sisotool(G)
#+end_src
#+end_exercice
In order to have the wanted Roll-off, two integrators are used, a lead is also added to have sufficient phase margin.
The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]].
#+begin_src matlab
K = 6e4 * ... % Gain
1/(s^2) * ... % Double Integrator
(1 + s/111)/(1 + s/888); % Lead
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-5, 1e1])
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-360, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/loop_gain_manual_afm.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:loop_gain_manual_afm
#+caption: Bode Plot of the obtained Loop Gain $L(s) = G(s) K(s)$
#+RESULTS:
[[file:figs/loop_gain_manual_afm.png]]
And we can verify that we have the wanted stability margins:
#+begin_src matlab :results output replace
[Gm, Pm, ~, Wc] = margin(G*K)
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin [dB]', 'Phase Margin [deg]', 'Crossover [Hz]'}, {'Value'}, ' %.1f ');
#+end_src
#+RESULTS:
| | Value |
|--------------------+-------|
| Gain Margin [dB] | 7.2 |
| Phase Margin [deg] | 48.1 |
| Crossover [Hz] | 50.7 |
** $\mathcal{H}_\infty$ Loop Shaping Synthesis
The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Synthesis.
Even though we will not go into details, we will provide one example.
Using Matlab, the $\mathcal{H}_\infty$ synthesis of a controller based on the wanted open loop shape can be performed using the =loopsyn= command:
#+begin_src matlab :eval no
K = loopsyn(G, Gd);
#+end_src
where:
- =G= is the (LTI) plant
- =Gd= is the wanted loop shape
- =K= is the synthesize controller
#+begin_seealso
Matlab documentation of =loopsyn= ([[https://www.mathworks.com/help/robust/ref/loopsyn.html][link]]).
#+end_seealso
** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
Let's re-use the previous plant.
Translate the specification into the wanted shape of the open loop gain.
#+begin_src matlab
G = tf(16,[1 0.16 16]);
Gd = 3.7e4*1/s*(1 + s/2/pi/20)/(1 + s/2/pi/220)*1/(s + s/2/pi/500);
#+end_src
#+begin_src matlab :exports none
bodeFig({Gd}, struct('phase', true))
#+end_src
#+begin_src matlab
[K,CL,GAM,INFO] = loopsyn(G, Gd);
#+end_src
#+begin_src matlab
bodeFig({K})
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz'))), 'k--');
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz')))*GAM, 'k-.');
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz')))/GAM, 'k-.');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-5, 1e1])
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-360, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
* The $\mathcal{H}_\infty$ Norm
#+begin_definition