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<h1 class="title">A brief and practical introduction to \(\mathcal{H}_\infty\) Control</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org51f6d5a">1. Introduction to the Control Methodology - Model Based Control</a>
<ul>
<li><a href="#orgec5f069">1.1. Model Based Control - Methodology</a></li>
<li><a href="#org3d1aaa3">1.2. Some Background: From Classical Control to Robust Control</a></li>
<li><a href="#orgaea0d8a">1.3. Example System</a></li>
</ul>
</li>
<li><a href="#orga20b0f9">2. Classical Open Loop Shaping</a>
<ul>
<li><a href="#orgfe3c688">2.1. Introduction to Loop Shaping</a></li>
<li><a href="#org69bd43a">2.2. Example of Open Loop Shaping</a></li>
<li><a href="#org8cfcaf9">2.3. \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
<li><a href="#orgf4b401b">2.4. Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
</ul>
</li>
<li><a href="#org0c67a17">3. First Steps in the \(\mathcal{H}_\infty\) world</a>
<ul>
<li><a href="#orgece6aff">3.1. The \(\mathcal{H}_\infty\) Norm</a></li>
<li><a href="#orgf113a48">3.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#orgf9f471c">3.3. The Generalized Plant</a></li>
<li><a href="#org4161a55">3.4. The General Synthesis Problem Formulation</a></li>
<li><a href="#orgf3e47de">3.5. From a Classical Feedback Architecture to a Generalized Plant</a></li>
</ul>
</li>
<li><a href="#org715c56d">4. Modern Interpretation of the Control Specifications</a>
<ul>
<li><a href="#orgd0340bf">4.1. Introduction</a></li>
<li><a href="#org0a9eee5">4.2. Closed Loop Transfer Functions</a></li>
<li><a href="#orgeceb831">4.3. Sensitivity Function</a></li>
<li><a href="#orgb1dfd8e">4.4. Robustness: Module Margin</a></li>
<li><a href="#org5ffd8b0">4.5. How to <b>Shape</b> transfer function? Using of Weighting Functions!</a></li>
<li><a href="#org57fbb94">4.6. Design of Weighting Functions</a></li>
<li><a href="#orge27bc96">4.7. Sensitivity Function Shaping - Example</a></li>
<li><a href="#org5643309">4.8. Complementary Sensitivity Function</a></li>
<li><a href="#orgff0dad8">4.9. Summary</a></li>
</ul>
</li>
<li><a href="#orgb9b8856">5. \(\mathcal{H}_\infty\) Mixed-Sensitivity Synthesis</a>
<ul>
<li><a href="#orgaba43d8">5.1. Problem</a></li>
<li><a href="#orgca1e14b">5.2. Typical Procedure</a></li>
<li><a href="#org0fb883d">5.3. Step 1 - Shaping of the Sensitivity Function</a></li>
<li><a href="#org4b74053">5.4. Step 2 - Shaping of</a></li>
<li><a href="#org4acb60b">5.5. General Configuration for various shaping</a></li>
</ul>
</li>
<li><a href="#org49ed4be">6. Conclusion</a></li>
<li><a href="#org0386aa4">7. Resources</a></li>
</ul>
</div>
</div>
<div id="outline-container-org51f6d5a" class="outline-2">
<h2 id="org51f6d5a"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org9888131"></a>
</p>
</div>
<div id="outline-container-orgec5f069" class="outline-3">
<h3 id="orgec5f069"><span class="section-number-3">1.1</span> Model Based Control - Methodology</h3>
<div class="outline-text-3" id="text-1-1">
<p>
<a id="org863a570"></a>
</p>
<p>
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#orgcc52183">1</a>.
It consists of three steps:
</p>
<ol class="org-ol">
<li><b>Identification or modeling</b>: \(\Longrightarrow\) mathematical model</li>
<li><b>Translate the specifications into mathematical criteria</b>:
<ul class="org-ul">
<li><span class="underline">Specifications</span>: Response Time, Noise Rejection, Maximum input amplitude, Robustness, &#x2026;</li>
<li><span class="underline">Mathematical Criteria</span>: Cost Function, Shape of TF</li>
</ul></li>
<li><b>Synthesis</b>: research of \(K\) that satisfies the specifications for the model of the system</li>
</ol>
<div id="orgcc52183" 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>
</div>
<p>
In this document, we will mainly focus on steps 2 and 3.
</p>
</div>
</div>
<div id="outline-container-org3d1aaa3" class="outline-3">
<h3 id="org3d1aaa3"><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>
<a id="orgbe25e33"></a>
</p>
<table id="org571232c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Table summurazing the main differences between classical, modern and robust control</caption>
<colgroup>
<col class="org-left" />
<col class="org-center" />
<col class="org-center" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-center"><b>Classical Control</b></th>
<th scope="col" class="org-center"><b>Modern Control</b></th>
<th scope="col" class="org-center"><b>Robust Control</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><b>Date</b></td>
<td class="org-center">1930-</td>
<td class="org-center">1960-</td>
<td class="org-center">1980-</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Tools</b></td>
<td class="org-center">Transfer Functions</td>
<td class="org-center">State Space formulation</td>
<td class="org-center">Systems and Signals Norms (\(\mathcal{H}_\infty\), \(\mathcal{H}_2\) Norms)</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">Nyquist Plots</td>
<td class="org-center">Riccati Equations</td>
<td class="org-center">Closed Loop Transfer Functions</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">Bode Plots</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">Open/Closed Loop Shaping</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">Phase and Gain margins</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">Weighting Functions</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">Disk margin</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Control Architectures</b></td>
<td class="org-center">Proportional, Integral, Derivative</td>
<td class="org-center">Full State Feedback</td>
<td class="org-center">General Control Configuration</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">Leads, Lags</td>
<td class="org-center">LQR, LQG</td>
<td class="org-center">&#xa0;</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">Kalman Filters</td>
<td class="org-center">&#xa0;</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Advantages</b></td>
<td class="org-center">Study Stability</td>
<td class="org-center">Automatic Synthesis</td>
<td class="org-center">Automatic Synthesis</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">Simple</td>
<td class="org-center">MIMO</td>
<td class="org-center">MIMO</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">Natural</td>
<td class="org-center">Optimization Problem</td>
<td class="org-center">Optimization Problem</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">Guaranteed Robustness</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">Easy specification of performances</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Disadvantages</b></td>
<td class="org-center">Manual Method</td>
<td class="org-center">No Guaranteed Robustness</td>
<td class="org-center">Required knowledge of specific tools</td>
</tr>
<tr>
<td class="org-left">&#xa0;</td>
<td class="org-center">Only SISO</td>
<td class="org-center">Difficult Rejection of Perturbations</td>
<td class="org-center">Need a reasonably good model of the system</td>
</tr>
</tbody>
</table>
<div id="orgda7aec2" class="figure">
<p><img src="figs/robustness_performance.png" alt="robustness_performance.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Comparison of the performance and robustness of classical control methods, modern control methods and robust control methods. The required information on the plant to succesfuly apply each of the control methods are indicated by the colors.</p>
</div>
</div>
</div>
<div id="outline-container-orgaea0d8a" class="outline-3">
<h3 id="orgaea0d8a"><span class="section-number-3">1.3</span> Example System</h3>
<div class="outline-text-3" id="text-1-3">
<p>
<a id="org70c9196"></a>
</p>
<p>
Let&rsquo;s consider the model shown in Figure <a href="#orgad0848c">3</a>.
It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor.
The notations used are listed in Table <a href="#org7ff33c4">2</a>.
</p>
<div id="orgad0848c" 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 3: </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. A feedback controller \(K(s)\) is added to position / isolate the payload.</p>
</div>
<table id="org7ff33c4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</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">\(10\)</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">\(10^6\)</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">\(400\)</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="orgcedc5e1">
<p>
Derive the following open-loop transfer functions:
</p>
\begin{align}
G(s) &= \frac{y}{u} \\
G_d(s) &= \frac{y}{d}
\end{align}
<details><summary>Hint</summary>
<p>
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>
</details>
<details><summary>Results</summary>
\begin{align}
G(s) &= \frac{1}{m s^2 + cs + k} \\
G_d(s) &= \frac{cs + k}{m s^2 + cs + k}
\end{align}
</details>
</div>
<p>
Hi Musa,
Thank you very much for sharing this awesome package.
For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by &lt;details&gt; blocks.
</p>
<p>
For now, I am manually adding #+HTML: &lt;details&gt;&lt;summary&gt;Code&lt;/summary&gt; and #+HTML: &lt;/details&gt; around the source blocks I want to hide&#x2026;
This is a very simple solution, but not so elegent nor practical.
</p>
<p>
Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?
</p>
<p>
Similarly, I would love to be able to export a &lt;span&gt; block with the name of the file corresponding to the source block.
For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting!
</p>
<p>
Thanks in advance
</p>
<p>
Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure <a href="#orgad0848c">3</a> into a classical feedback form as shown in Figure <a href="#orgd2cb22a">7</a>.
</p>
<div id="orgcc2d43f" class="figure">
<p><img src="figs/classical_feedback_test_system.png" alt="classical_feedback_test_system.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Block diagram corresponding to the example system</p>
</div>
<p>
Let&rsquo;s define the system parameters on Matlab.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="linenr">1: </span>k = 1e6; <span class="org-comment">% Stiffness [N/m]</span>
<span class="linenr">2: </span>c = 4e2; <span class="org-comment">% Damping [N/(m/s)]</span>
<span class="linenr">3: </span>m = 10; <span class="org-comment">% Mass [kg]</span>
</pre>
</div>
<p>
And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown in Figures <a href="#org8fbda48">5</a> and <a href="#orgad42b8b">6</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="linenr">4: </span>G = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c<span class="org-type">*</span>s <span class="org-type">+</span> k); <span class="org-comment">% Plant</span>
<span class="linenr">5: </span>Gd = (c<span class="org-type">*</span>s <span class="org-type">+</span> k)<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c<span class="org-type">*</span>s <span class="org-type">+</span> k); <span class="org-comment">% Disturbance</span>
</pre>
</div>
<div id="org8fbda48" 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>
<div id="orgad42b8b" class="figure">
<p><img src="figs/bode_plot_example_Gd.png" alt="bode_plot_example_Gd.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Magnitude of the disturbance transfer function \(G_d(s)\)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orga20b0f9" class="outline-2">
<h2 id="orga20b0f9"><span class="section-number-2">2</span> Classical Open Loop Shaping</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orgb1fe32e"></a>
</p>
</div>
<div id="outline-container-orgfe3c688" class="outline-3">
<h3 id="orgfe3c688"><span class="section-number-3">2.1</span> Introduction to Loop Shaping</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="orge774dc4"></a>
</p>
<div class="definition" id="org2f3f337">
<p>
<b>Loop Shaping</b> refers to a design procedure that involves explicitly shaping the magnitude of the <b>Loop Transfer Function</b> \(L(s)\).
</p>
</div>
<div class="definition" id="orgb83578b">
<p>
The <b>Loop Gain</b> \(L(s)\) usually refers to as the product of the controller and the plant (&ldquo;Gain around the loop&rdquo;, see Figure <a href="#orgd2cb22a">7</a>):
</p>
\begin{equation}
L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
\end{equation}
<div id="orgd2cb22a" class="figure">
<p><img src="figs/open_loop_shaping.png" alt="open_loop_shaping.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Classical Feedback Architecture</p>
</div>
</div>
<p>
This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain \(L(s)\) such as:
</p>
<ul class="org-ul">
<li><b>Performance</b>: \(L\) large</li>
<li><b>Good disturbance rejection</b>: \(L\) large</li>
<li><b>Limitation of measurement noise on plant output</b>: \(L\) small</li>
<li><b>Small magnitude of input signal</b>: \(K\) and \(L\) small</li>
<li><b>Nominal stability</b>: \(L\) small (RHP zeros and time delays)</li>
<li><b>Robust stability</b>: \(L\) small (neglected dynamics)</li>
</ul>
<p>
The Open Loop shape is usually done manually has the loop gain \(L(s)\) depends linearly on \(K(s)\) \eqref{eq:loop_gain}.
</p>
<p>
\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that \(L(s)\) has the wanted shape (an example is shown in Figure <a href="#org863f5d7">8</a>).
</p>
<div id="org863f5d7" class="figure">
<p><img src="figs/open_loop_shaping_shape.png" alt="open_loop_shaping_shape.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Typical Wanted Shape for the Loop Gain \(L(s)\)</p>
</div>
</div>
</div>
<div id="outline-container-org69bd43a" class="outline-3">
<h3 id="org69bd43a"><span class="section-number-3">2.2</span> Example of Open Loop Shaping</h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="org359f4cc"></a>
</p>
<div class="exampl" id="orgb4992f1">
<p>
Let&rsquo;s take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
</p>
<ul class="org-ul">
<li><b>Performance</b>: Bandwidth of approximately 10Hz</li>
<li><b>Noise Attenuation</b>: Roll-off of -40dB/decade past 30Hz</li>
<li><b>Robustness</b>: Gain margin &gt; 3dB and Phase margin &gt; 30 deg</li>
</ul>
</div>
<div class="exercice" id="org2cf0e3e">
<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="#org91530d7">9</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = 14e8 <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>(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>40) <span class="org-type">*</span> ...<span class="org-comment"> % Low Pass Filter</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>10<span class="org-type">/</span>sqrt(8)))<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>10<span class="org-type">*</span>sqrt(8))); <span class="org-comment">% Lead</span>
</pre>
</div>
<div id="org91530d7" class="figure">
<p><img src="figs/loop_gain_manual_afm.png" alt="loop_gain_manual_afm.png" />
</p>
<p><span class="figure-number">Figure 9: </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">Requirements</th>
<th scope="col" class="org-right">Manual Method</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Gain Margin \(> 3\) [dB]</td>
<td class="org-right">3.1</td>
</tr>
<tr>
<td class="org-left">Phase Margin \(> 30\) [deg]</td>
<td class="org-right">35.4</td>
</tr>
<tr>
<td class="org-left">Crossover \(\approx 10\) [Hz]</td>
<td class="org-right">10.1</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org8cfcaf9" class="outline-3">
<h3 id="org8cfcaf9"><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>
<a id="org841deda"></a>
</p>
<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 \(\mathcal{H}_\infty\) Loop Shaping Synthesis 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="org21beb6c">
<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-orgf4b401b" class="outline-3">
<h3 id="orgf4b401b"><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>
<a id="orgc8bcd75"></a>
</p>
<p>
Let&rsquo;s reuse the previous plant.
</p>
<p>
Translate the specification into the wanted shape of the open loop gain.
</p>
<ul class="org-ul">
<li><b>Performance</b>: Bandwidth of approximately 10Hz: \(|L_w(j2 \pi 10)| = 1\)</li>
<li><b>Noise Attenuation</b>: Roll-off of -40dB/decade past 30Hz</li>
<li><b>Robustness</b>: Gain margin &gt; 3dB and Phase margin &gt; 30 deg</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">Lw = 2.3e3 <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-comment"> % Double Integrator</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>10<span class="org-type">/</span>sqrt(3)))<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>10<span class="org-type">*</span>sqrt(3))); <span class="org-comment">% Lead</span>
</pre>
</div>
<p>
The \(\mathcal{H}_\infty\) optimal open loop shaping synthesis is performed using the <code>loopsyn</code> command:
</p>
<div class="org-src-container">
<pre class="src src-matlab">[K, <span class="org-type">~</span>, GAM] = loopsyn(G, Lw);
</pre>
</div>
<p>
The Bode plot of the obtained controller is shown in Figure <a href="#org867906f">10</a>.
</p>
<div class="important" id="org70a9456">
<p>
It is always important to analyze the controller after the synthesis is performed.
</p>
<p>
In the end, a synthesize controller is just a combination of low pass filters, high pass filters, notches, leads, etc.
</p>
</div>
<p>
Let&rsquo;s briefly analyze this controller:
</p>
<ul class="org-ul">
<li>two integrators are used at low frequency to have the wanted low frequency high gain</li>
<li>a lead is added centered with the crossover frequency to increase the phase margin</li>
<li>a notch is added at the resonance of the plant to increase the gain margin (this is very typical of \(\mathcal{H}_\infty\) controllers, and can be an issue, more info on that latter)</li>
</ul>
<div id="org867906f" class="figure">
<p><img src="figs/open_loop_shaping_hinf_K.png" alt="open_loop_shaping_hinf_K.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Obtained controller \(K\) using the open-loop \(\mathcal{H}_\infty\) shaping</p>
</div>
<p>
The obtained Loop Gain is shown in Figure <a href="#orga97b54d">11</a>.
</p>
<div id="orga97b54d" class="figure">
<p><img src="figs/open_loop_shaping_hinf_L.png" alt="open_loop_shaping_hinf_L.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Obtained Open Loop Gain \(L(s) = G(s) K(s)\) and comparison with the wanted Loop gain \(L_w\)</p>
</div>
<p>
Let&rsquo;s now compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table <a href="#org1f809d0">3</a>.
</p>
<table id="org1f809d0" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Comparison of the characteristics obtained with the two methods</caption>
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Specifications</th>
<th scope="col" class="org-right">Manual Method</th>
<th scope="col" class="org-right">\(\mathcal{H}_\infty\) Method</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Gain Margin \(> 3\) [dB]</td>
<td class="org-right">3.1</td>
<td class="org-right">31.7</td>
</tr>
<tr>
<td class="org-left">Phase Margin \(> 30\) [deg]</td>
<td class="org-right">35.4</td>
<td class="org-right">54.7</td>
</tr>
<tr>
<td class="org-left">Crossover \(\approx 10\) [Hz]</td>
<td class="org-right">10.1</td>
<td class="org-right">9.9</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-org0c67a17" class="outline-2">
<h2 id="org0c67a17"><span class="section-number-2">3</span> First Steps in the \(\mathcal{H}_\infty\) world</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="orge61e7b8"></a>
</p>
</div>
<div id="outline-container-orgece6aff" class="outline-3">
<h3 id="orgece6aff"><span class="section-number-3">3.1</span> The \(\mathcal{H}_\infty\) Norm</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="orga53489a"></a>
</p>
<div class="definition" id="org8fedf4d">
<p>
The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
</p>
\begin{equation}
\|G(s)\|_\infty = \max_\omega \bar{\sigma}\big( G(j\omega) \big)
\end{equation}
<p>
For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a function of frequency:
</p>
\begin{equation}
\|G(s)\|_\infty = \max_{\omega} |G(j\omega)| \label{eq:hinf_norm_siso}
\end{equation}
</div>
<div class="exampl" id="org6a84e6f">
<p>
Let&rsquo;s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) using the <code>hinfnorm</code> function:
</p>
<div class="org-src-container">
<pre class="src src-matlab">hinfnorm(G)
</pre>
</div>
<pre class="example">
7.9216e-06
</pre>
<p>
We can see that the \(\mathcal{H}_\infty\) norm of \(G(s)\) does corresponds to the peak value of \(|G(j\omega)|\) as a function of frequency as shown in Figure <a href="#org7821c4b">12</a>.
</p>
<div id="org7821c4b" class="figure">
<p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgf113a48" class="outline-3">
<h3 id="orgf113a48"><span class="section-number-3">3.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="org0c0df60"></a>
</p>
<div class="definition" id="org84c7b9a">
<p>
\(\mathcal{H}_\infty\) synthesis is a method that uses an <b>algorithm</b> (LMI optimization, Riccati equation) to find a controller that stabilize the system and that <b>minimizes</b> the \(\mathcal{H}_\infty\) norms of defined transfer functions.
</p>
</div>
<p>
Why optimizing the \(\mathcal{H}_\infty\) norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the \(\mathcal{H}_\infty\) norm of transfer functions.
</p>
<p>
Then applying the \(\mathcal{H}_\infty\) synthesis to a plant, the engineer work usually consists of the following steps
</p>
<ol class="org-ol">
<li>Write the problem as standard \(\mathcal{H}_\infty\) problem</li>
<li>Translate the specifications as \(\mathcal{H}_\infty\) norms of transfer functions</li>
<li>Make the synthesis and analyze the obtain controller</li>
<li>Reduce the order of the controller for implementation</li>
</ol>
<p>
Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
</p>
<ul class="org-ul">
<li>Traditional \(\mathcal{H}_\infty\) Synthesis (<code>hinfsyn</code> <a href="https://www.mathworks.com/help/robust/ref/hinfsyn.html">doc</a>)</li>
<li>Open Loop Shaping \(\mathcal{H}_\infty\) Synthesis (<code>loopsyn</code> <a href="https://www.mathworks.com/help/robust/ref/loopsyn.html">doc</a>)</li>
<li>Mixed Sensitivity Loop Shaping (<code>mixsyn</code> <a href="https://www.mathworks.com/help/robust/ref/lti.mixsyn.html">doc</a>)</li>
<li>Fixed-Structure \(\mathcal{H}_\infty\) Synthesis (<code>hinfstruct</code> <a href="https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html">doc</a>)</li>
<li>Signal Based \(\mathcal{H}_\infty\) Synthesis</li>
</ul>
</div>
</div>
<div id="outline-container-orgf9f471c" class="outline-3">
<h3 id="orgf9f471c"><span class="section-number-3">3.3</span> The Generalized Plant</h3>
<div class="outline-text-3" id="text-3-3">
<p>
<a id="orgf417ccf"></a>
</p>
<p>
The first step when applying the \(\mathcal{H}_\infty\) synthesis is usually to write the problem as a standard \(\mathcal{H}_\infty\) problem.
This consist of deriving the <b>Generalized Plant</b> for the current problem.
It makes things much easier for the following steps.
</p>
<p>
The generalized plant, usually noted \(P(s)\), is shown in Figure <a href="#org4737336">13</a>.
It has two inputs and two outputs (both could contains many signals).
The meaning of the inputs and outputs are summarized in Table <a href="#org1baeacb">4</a>.
</p>
<p>
Note that this generalized plant is as its name implies, quite <i>general</i>.
It can indeed represent feedback as well as feedforward control architectures.
</p>
\begin{equation}
\begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix} = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w \\ u \end{bmatrix}
\end{equation}
<div id="org4737336" class="figure">
<p><img src="figs/general_plant.png" alt="general_plant.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Inputs and Outputs of the generalized Plant</p>
</div>
<div class="important" id="org5035437">
<table id="org1baeacb" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Notations for the general configuration</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Notation</th>
<th scope="col" class="org-left">Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(P\)</td>
<td class="org-left">Generalized plant model</td>
</tr>
<tr>
<td class="org-left">\(w\)</td>
<td class="org-left">Exogenous inputs: commands, disturbances, noise</td>
</tr>
<tr>
<td class="org-left">\(z\)</td>
<td class="org-left">Exogenous outputs: signals to be minimized</td>
</tr>
<tr>
<td class="org-left">\(v\)</td>
<td class="org-left">Controller inputs: measurements</td>
</tr>
<tr>
<td class="org-left">\(u\)</td>
<td class="org-left">Control signals</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-org4161a55" class="outline-3">
<h3 id="org4161a55"><span class="section-number-3">3.4</span> The General Synthesis Problem Formulation</h3>
<div class="outline-text-3" id="text-3-4">
<p>
<a id="org69c48d4"></a>
</p>
<p>
Once the generalized plant is obtained, the \(\mathcal{H}_\infty\) synthesis problem can be stated as follows:
</p>
<div class="important" id="org6f60b56">
<dl class="org-dl">
<dt>\(\mathcal{H}_\infty\) Synthesis applied on the generalized plant</dt><dd></dd>
</dl>
<p>
Find a stabilizing controller \(K\) that, using the sensed output \(v\), generates a control signal \(u\) such that the \(\mathcal{H}_\infty\) norm of the closed-loop transfer function from \(w\) to \(z\) is minimized.
</p>
<p>
After \(K\) is found, the system is <i>robustified</i> by adjusting the response around the unity gain frequency to increase stability margins.
</p>
</div>
<div id="org7909109" class="figure">
<p><img src="figs/general_control_names.png" alt="general_control_names.png" />
</p>
<p><span class="figure-number">Figure 14: </span>General Control Configuration</p>
</div>
<p>
Note that the closed-loop transfer function from \(w\) to \(z\) is:
</p>
\begin{equation}
\frac{z}{w} = P_{11} + P_{12} K \big( I - P_{22} K \big)^{-1} P_{21} \triangleq F_l(P, K)
\end{equation}
<p>
Using Matlab, the \(\mathcal{H}_\infty\) Synthesis applied on a Generalized plant can be applied using the <code>hinfsyn</code> command (<a href="https://www.mathworks.com/help/robust/ref/hinfsyn.html">documentation</a>):
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = hinfsyn(P, nmeas, ncont);
</pre>
</div>
<p>
where:
</p>
<ul class="org-ul">
<li><code>P</code> is the generalized plant transfer function matrix</li>
<li><code>nmeas</code> is the number of sensed output (size of \(v\))</li>
<li><code>ncont</code> is the number of control signals (size of \(u\))</li>
<li><code>K</code> obtained controller that minimized the \(\mathcal{H}_\infty\) norm from \(w\) to \(z\)</li>
</ul>
</div>
</div>
<div id="outline-container-orgf3e47de" class="outline-3">
<h3 id="orgf3e47de"><span class="section-number-3">3.5</span> From a Classical Feedback Architecture to a Generalized Plant</h3>
<div class="outline-text-3" id="text-3-5">
<p>
<a id="org03c8edd"></a>
</p>
<p>
The procedure to convert a typical control architecture as the one shown in Figure <a href="#orgc795e53">15</a> to a generalized Plant is as follows:
</p>
<ol class="org-ol">
<li>Define signals (\(w\), \(z\), \(u\) and \(v\)) of the generalized plant</li>
<li>Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration</li>
</ol>
<div class="exercice" id="orgf8568c7">
<p>
Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgc795e53">15</a>
</p>
<div id="orgc795e53" class="figure">
<p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Classical Feedback Control Architecture (Tracking)</p>
</div>
<details><summary>Hint</summary>
<p>
First, define the signals of the generalized plant:
</p>
<ul class="org-ul">
<li>Exogenous inputs: \(w = r\)</li>
<li>Signals to be minimized: \(z_1 = \epsilon\), \(z_2 = u\)</li>
<li>Control signals: \(v = y\)</li>
<li>Control inputs: \(u\)</li>
</ul>
<p>
Then, Remove \(K\) and rearrange the inputs and outputs.
</p>
</details>
<details><summary>Answer</summary>
<p>
The obtained generalized plant shown in Figure <a href="#org988ad81">16</a>.
</p>
<div id="org988ad81" class="figure">
<p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p>
</div>
</details>
</div>
<p>
Using Matlab, the generalized plant can be defined as follows:
</p>
<div class="org-src-container">
<pre class="src src-matlab">P = [1 <span class="org-type">-</span>G;
0 1;
1 <span class="org-type">-</span>G]
P.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'u'</span>};
P.OutputName = {<span class="org-string">'e'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>};
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org715c56d" class="outline-2">
<h2 id="org715c56d"><span class="section-number-2">4</span> Modern Interpretation of the Control Specifications</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org9ad723c"></a>
</p>
</div>
<div id="outline-container-orgd0340bf" class="outline-3">
<h3 id="orgd0340bf"><span class="section-number-3">4.1</span> Introduction</h3>
<div class="outline-text-3" id="text-4-1">
<p>
As shown in Section <a href="#orgb1fe32e">2</a>, the loop gain \(L(s) = G(s) K(s)\) is a useful and easy tool for the design of controllers by hand.
</p>
<p>
It is very easy to shape as it depends linearly on \(K(s)\).
Moreover, it gives information on important quantities such as the stability margins and the control bandwidth.
</p>
<p>
However, the loop gain \(L(s)\) does <b>not</b> directly give the performances of the closed-loop system.
The closed loop system behavior is determined by the <b>closed-loop transfer functions</b>.
</p>
<p>
If we consider the feedback system shown in Figure <a href="#org8e440ea">17</a>, we can link to the following specifications to closed-loop transfer functions
</p>
<ul class="org-ul">
<li><b>Reference tracking</b> (Overshoot, Static error, Settling time, &#x2026;)
<ul class="org-ul">
<li>From \(r\) to \(\epsilon\)</li>
</ul></li>
<li><b>Disturbances rejection</b>
<ul class="org-ul">
<li>From \(d\) to \(y\)</li>
</ul></li>
<li><b>Measurement noise filtering</b>
<ul class="org-ul">
<li>From \(n\) to \(y\)</li>
</ul></li>
<li><b>Small command amplitude</b>
<ul class="org-ul">
<li>From \(n, r, d\) to \(u\)</li>
</ul></li>
<li><b>Stability</b>
<ul class="org-ul">
<li>All closed-loop transfer functions must be stable</li>
</ul></li>
<li><b>Robustness</b> (stability margins)
<ul class="org-ul">
<li>Module margin (see Section <a href="#orga56149f">4.4</a>)</li>
</ul></li>
</ul>
<div id="org8e440ea" class="figure">
<p><img src="figs/gang_of_four_feedback.png" alt="gang_of_four_feedback.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Simple Feedback Architecture</p>
</div>
</div>
</div>
<div id="outline-container-org0a9eee5" class="outline-3">
<h3 id="org0a9eee5"><span class="section-number-3">4.2</span> Closed Loop Transfer Functions</h3>
<div class="outline-text-3" id="text-4-2">
<p>
<a id="orge8b7301"></a>
</p>
<p>
As the performances of a controlled system depend on the <b>closed</b> loop transfer functions, it is very important to derive these closed-loop transfer functions as a function of the plant \(G(s)\) and controller \(K(s)\).
</p>
<div class="exercice" id="org7b6ac0e">
<p>
Write the output signals \([\epsilon, u, y]\) as a function of the systems \(K(s), G(s)\) and of the input signals \([r, d, n]\) as shown in Figure <a href="#org8e440ea">17</a>.
</p>
<details><summary>Hint</summary>
<p>
Take one of the output (e.g. \(y\)), and write it as a function of the inputs \([d, r, n]\) going step by step around the loop:
</p>
\begin{aligned}
y &= G u \\
&= G (d + K \epsilon) \\
&= G \big(d + K (r - n - y) \big) \\
&= G d + GK r - GK n - GK y
\end{aligned}
<p>
Isolate \(y\) at the right hand side, and finally obtain:
\[ y = \frac{GK}{1+ GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \]
</p>
<p>
Do the same procedure for \(u\) and \(\epsilon\)
</p>
</details>
<details><summary>Anwser</summary>
<p>
The following equations should be obtained:
</p>
\begin{align}
y &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\
\epsilon &= \frac{1 }{1 + GK} r - \frac{G}{1 + GK} d - \frac{G }{1 + GK} n \\
u &= \frac{K }{1 + GK} r - \frac{1}{1 + GK} d - \frac{K }{1 + GK} n
\end{align}
</details>
</div>
<div class="important" id="org2311e7f">
<p>
We can see that they are 4 different transfer functions describing the behavior of the system in Figure <a href="#org8e440ea">17</a>.
These called the <b>Gang of Four</b>:
</p>
\begin{align}
S &= \frac{1 }{1 + GK}, \quad \text{the sensitivity function} \\
T &= \frac{GK}{1 + GK}, \quad \text{the complementary sensitivity function} \\
GS &= \frac{G }{1 + GK}, \quad \text{the load disturbance sensitivity function} \\
KS &= \frac{K }{1 + GK}, \quad \text{the noise sensitivity function}
\end{align}
</div>
<div class="seealso" id="orgea740e1">
<p>
If a feedforward controller is included, a <b>Gang of Six</b> transfer functions can be defined.
More on that in this <a href="https://www.youtube.com/watch?v=b_8v8scghh8">short video</a>.
</p>
</div>
<p>
And we have:
</p>
\begin{align}
\epsilon &= S r - GS d - GS n \\
y &= T r + GS d - T n \\
u &= KS r - S d - KS n
\end{align}
<p>
Thus, for reference tracking, we want to shape the <i>closed-loop</i> transfer function from \(r\) to \(\epsilon\), that is the sensitivity function \(S(s)\).
Similarly, to reduce the effect of measurement noise \(n\) on the output \(y\), we want to act on the complementary sensitivity function \(T(s)\).
</p>
</div>
</div>
<div id="outline-container-orgeceb831" class="outline-3">
<h3 id="orgeceb831"><span class="section-number-3">4.3</span> Sensitivity Function</h3>
<div class="outline-text-3" id="text-4-3">
<p>
<a id="orga561a85"></a>
</p>
<div class="org-src-container">
<pre class="src src-matlab">K1 = 14e8 <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>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10<span class="org-type">/</span>sqrt(8)))<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>10<span class="org-type">*</span>sqrt(8))); <span class="org-comment">% Lead</span>
K2 = 1e8 <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>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1<span class="org-type">/</span>sqrt(8)))<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>1<span class="org-type">*</span>sqrt(8))); <span class="org-comment">% Lead</span>
K3 = 1e8 <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>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1<span class="org-type">/</span>sqrt(2)))<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>1<span class="org-type">*</span>sqrt(2))); <span class="org-comment">% Lead</span>
S1 = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> K1<span class="org-type">*</span>G);
S2 = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> K2<span class="org-type">*</span>G);
S3 = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> K3<span class="org-type">*</span>G);
T1 = K1<span class="org-type">*</span>G<span class="org-type">/</span>(1 <span class="org-type">+</span> K1<span class="org-type">*</span>G);
T2 = K2<span class="org-type">*</span>G<span class="org-type">/</span>(1 <span class="org-type">+</span> K2<span class="org-type">*</span>G);
T3 = K3<span class="org-type">*</span>G<span class="org-type">/</span>(1 <span class="org-type">+</span> K3<span class="org-type">*</span>G);
bodeFig({S1, S2, S3})
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(<span class="org-type">-</span>1, 2, 1000);
<span class="org-type">figure</span>;
tiledlayout(1, 2, <span class="org-string">'TileSpacing'</span>, <span class="org-string">'None'</span>, <span class="org-string">'Padding'</span>, <span class="org-string">'None'</span>);
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(S1, freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$L(s)$'</span>);
plot(freqs, abs(squeeze(freqresp(S2, freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$L_w(s)$'</span>);
plot(freqs, abs(squeeze(freqresp(S3, freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$'</span>);
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frquency [Hz]'</span>); ylabel(<span class="org-string">'Sensitivity Magnitude'</span>);
hold off;
ax2 = nexttile;
t = linspace(0, 1, 1000);
y1 = step(T1, t);
y2 = step(T2, t);
y3 = step(T3, t);
hold on;
plot(t, y1)
plot(t, y2)
plot(t, y3)
hold off
xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Step Response'</span>);
</pre>
</div>
<div id="org6f89001" class="figure">
<p><img src="figs/h-infinity-spec-S.png" alt="h-infinity-spec-S.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Typical wanted shape of the Sensitivity transfer function</p>
</div>
</div>
</div>
<div id="outline-container-orgb1dfd8e" class="outline-3">
<h3 id="orgb1dfd8e"><span class="section-number-3">4.4</span> Robustness: Module Margin</h3>
<div class="outline-text-3" id="text-4-4">
<p>
<a id="orga56149f"></a>
</p>
<ul class="org-ul">
<li class="off"><code>[&#xa0;]</code> Definition of Module margin</li>
<li class="off"><code>[&#xa0;]</code> Why it represents robustness</li>
<li class="off"><code>[&#xa0;]</code> Example</li>
</ul>
<p>
\[ M_S < 2 \Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o \]
</p>
</div>
</div>
<div id="outline-container-org5ffd8b0" class="outline-3">
<h3 id="org5ffd8b0"><span class="section-number-3">4.5</span> How to <b>Shape</b> transfer function? Using of Weighting Functions!</h3>
<div class="outline-text-3" id="text-4-5">
<p>
<a id="org228a87c"></a>
</p>
<ul class="org-ul">
<li class="off"><code>[&#xa0;]</code> Maybe put this section in Previous chapter</li>
</ul>
<p>
Let&rsquo;s say we want to shape the sensitivity transfer function corresponding to the transfer function from \(r\) to \(\epsilon\) of the control architecture shown in Figure <a href="#org411650e">19</a>.
</p>
<div id="org411650e" class="figure">
<p><img src="figs/loop_shaping_S_without_W.png" alt="loop_shaping_S_without_W.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Generalized Plant</p>
</div>
<p>
If the \(\mathcal{H}_\infty\) synthesis is directly applied on the generalized plant \(P(s)\) shown in Figure <a href="#org411650e">19</a>, if will minimize the \(\mathcal{H}_\infty\) norm of transfer function from \(r\) to \(\epsilon\) (the sensitivity transfer function).
</p>
<p>
However, as the \(\mathcal{H}_\infty\) norm is the maximum peak value of the transfer function&rsquo;s magnitude, it does not allow to <b>shape</b> the norm over all frequencies.
</p>
<p>
A <i>trick</i> is to include a <b>weighting function</b> in the generalized plant as shown in Figure <a href="#org448d5c3">20</a>.
Applying the \(\mathcal{H}_\infty\) synthesis to the <i>weighted</i> generalized plant \(\tilde{P}(s)\) (Figure <a href="#org448d5c3">20</a>) will generate a controller \(K(s)\) that minimizes the \(\mathcal{H}_\infty\) norm between \(r\) and \(\tilde{\epsilon}\):
</p>
\begin{align}
& \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1)\nonumber \\
\Leftrightarrow & \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\
\Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\
\Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping}
\end{align}
<div class="important" id="org0fccbdd">
<p>
As shown in Equation \eqref{eq:sensitivity_shaping}, the \(\mathcal{H}_\infty\) synthesis allows to <b>shape</b> the magnitude of the sensitivity transfer function.
Therefore, the choice of the weighting function \(W_s(s)\) is very important.
Its inverse magnitude will define the frequency dependent upper bound of the sensitivity transfer function magnitude.
</p>
</div>
<div id="org448d5c3" class="figure">
<p><img src="figs/loop_shaping_S_with_W.png" alt="loop_shaping_S_with_W.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Weighted Generalized Plant</p>
</div>
<p>
Once the weighting function is designed, it should be added to the generalized plant as shown in Figure <a href="#org448d5c3">20</a>.
</p>
<p>
The weighted generalized plant can be defined in Matlab by either re-defining all the inputs or by pre-multiplying the (non-weighted) generalized plant by a block-diagonal MIMO transfer function containing the weights for the outputs \(z\) and <code>1</code> for the outputs \(v\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Pw = [Ws <span class="org-type">-</span>Ws<span class="org-type">*</span>G;
1 <span class="org-type">-</span>G]
<span class="org-comment">% Alternative</span>
Pw = blkdiag(Ws, 1)<span class="org-type">*</span>P;
</pre>
</div>
</div>
</div>
<div id="outline-container-org57fbb94" class="outline-3">
<h3 id="org57fbb94"><span class="section-number-3">4.6</span> Design of Weighting Functions</h3>
<div class="outline-text-3" id="text-4-6">
<p>
<a id="org5da96f0"></a>
</p>
<p>
Weighting function used must be <b>proper</b>, <b>stable</b> and <b>minimum phase</b> transfer functions.
</p>
<dl class="org-dl">
<dt>proper</dt><dd>more poles than zeros, this implies \(\lim_{\omega \to \infty} |W(j\omega)| < \infty\)</dd>
<dt>stable</dt><dd>no poles in the right half plane</dd>
<dt>minimum phase</dt><dd>no zeros in the right half plane</dd>
</dl>
<p>
Matlab is providing the <code>makeweight</code> function that creates a first-order weights by specifying the low frequency gain, high frequency gain, and a gain at a specific frequency:
</p>
<div class="org-src-container">
<pre class="src src-matlab">W = makeweight(dcgain,[freq,mag],hfgain)
</pre>
</div>
<p>
with:
</p>
<ul class="org-ul">
<li><code>dcgain</code></li>
<li><code>freq</code></li>
<li><code>mag</code></li>
<li><code>hfgain</code></li>
</ul>
<div class="exampl" id="org0c5b88f">
<p>
The Matlab code below produces a weighting function with a magnitude shape shown in Figure <a href="#orge00a417">21</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ws = makeweight(1e2, [2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10, 1], 1<span class="org-type">/</span>2);
</pre>
</div>
<div id="orge00a417" class="figure">
<p><img src="figs/first_order_weight.png" alt="first_order_weight.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Obtained Magnitude of the Weighting Function</p>
</div>
</div>
<div class="seealso" id="orgf3d4410">
<p>
Quite often, higher orders weights are required.
</p>
<p>
In such case, the following formula can be used the design of these weights:
</p>
\begin{equation}
W(s) = \left( \frac{
\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
}{
\left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
}\right)^n \label{eq:weight_formula_advanced}
\end{equation}
<p>
The parameters permit to specify:
</p>
<ul class="org-ul">
<li>the low frequency gain: \(G_0 = lim_{\omega \to 0} |W(j\omega)|\)</li>
<li>the high frequency gain: \(G_\infty = lim_{\omega \to \infty} |W(j\omega)|\)</li>
<li>the absolute gain at \(\omega_0\): \(G_c = |W(j\omega_0)|\)</li>
<li>the absolute slope between high and low frequency: \(n\)</li>
</ul>
<p>
A Matlab function implementing Equation \eqref{eq:weight_formula_advanced} is shown below:
</p>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 1: </span>Matlab Function that can be used to generate Weighting functions</label><pre class="src src-matlab" id="orgc55218d"><span class="org-keyword">function</span> <span class="org-variable-name">[W]</span> = <span class="org-function-name">generateWeight</span>(<span class="org-variable-name">args</span>)
arguments
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
args.wc (1,1) double {mustBeNumeric, mustBePositive} = 2<span class="org-type">*</span><span class="org-constant">pi</span>
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> (args.Gc <span class="org-type">&lt;=</span> args.G0 <span class="org-type">&amp;&amp;</span> args.Gc <span class="org-type">&lt;=</span> args.G1) <span class="org-type">||</span> (args.Gc <span class="org-type">&gt;=</span> args.G0 <span class="org-type">&amp;&amp;</span> args.Gc <span class="org-type">&gt;=</span> args.G1)
eid = <span class="org-string">'value:range'</span>;
msg = <span class="org-string">'Gc must be between G0 and G1'</span>;
throwAsCaller(MException(eid,msg))
<span class="org-keyword">end</span>
s = zpk(<span class="org-string">'s'</span>);
W = (((1<span class="org-type">/</span>args.wc)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n))<span class="org-type">/</span>(1<span class="org-type">-</span>(args.Gc<span class="org-type">/</span>args.G1)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n)))<span class="org-type">*</span>s <span class="org-type">+</span> (args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n))<span class="org-type">/</span>((1<span class="org-type">/</span>args.G1)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n)<span class="org-type">*</span>(1<span class="org-type">/</span>args.wc)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n))<span class="org-type">/</span>(1<span class="org-type">-</span>(args.Gc<span class="org-type">/</span>args.G1)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n)))<span class="org-type">^</span>args.n;
<span class="org-keyword">end</span>
</pre>
</div>
<p>
Let&rsquo;s use this function to generate three weights with the same high and low frequency gains, but but different slopes.
</p>
<div class="org-src-container">
<pre class="src src-matlab">W1 = generateWeight(<span class="org-string">'G0'</span>, 1e2, <span class="org-string">'G1'</span>, 1<span class="org-type">/</span>2, <span class="org-string">'Gc'</span>, 1, <span class="org-string">'wc'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10, <span class="org-string">'n'</span>, 1);
W2 = generateWeight(<span class="org-string">'G0'</span>, 1e2, <span class="org-string">'G1'</span>, 1<span class="org-type">/</span>2, <span class="org-string">'Gc'</span>, 1, <span class="org-string">'wc'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10, <span class="org-string">'n'</span>, 2);
W3 = generateWeight(<span class="org-string">'G0'</span>, 1e2, <span class="org-string">'G1'</span>, 1<span class="org-type">/</span>2, <span class="org-string">'Gc'</span>, 1, <span class="org-string">'wc'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10, <span class="org-string">'n'</span>, 3);
</pre>
</div>
<p>
The obtained shapes are shown in Figure <a href="#org397f75d">22</a>.
</p>
<div id="org397f75d" class="figure">
<p><img src="figs/high_order_weight.png" alt="high_order_weight.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Higher order weights using Equation \eqref{eq:weight_formula_advanced}</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orge27bc96" class="outline-3">
<h3 id="orge27bc96"><span class="section-number-3">4.7</span> Sensitivity Function Shaping - Example</h3>
<div class="outline-text-3" id="text-4-7">
<p>
<a id="org05357ed"></a>
</p>
<ul class="org-ul">
<li>Robustness: Module margin &gt; 2 (\(\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o\))</li>
<li>Bandwidth:</li>
<li>Slope of -2</li>
</ul>
<p>
First, the weighting functions is generated.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ws = generateWeight(<span class="org-string">'G0'</span>, 1e3, <span class="org-string">'G1'</span>, 1<span class="org-type">/</span>2, <span class="org-string">'Gc'</span>, 1, <span class="org-string">'wc'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10, <span class="org-string">'n'</span>, 2);
</pre>
</div>
<p>
It is then added to the generalized plant.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Pw = blkdiag(Ws, 1)<span class="org-type">*</span>P;
</pre>
</div>
<p>
And the \(\mathcal{H}_\infty\) synthesis is performed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = hinfsyn(Pw, 1, 1, <span class="org-string">'Display'</span>, <span class="org-string">'on'</span>);
</pre>
</div>
<pre class="example" id="orgcc06779">
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
Test bounds: 0.5 &lt;= gamma &lt;= 0.51
gamma X&gt;=0 Y&gt;=0 rho(XY)&lt;1 p/f
5.05e-01 0.0e+00 0.0e+00 4.497e-28 p
Limiting gains...
5.05e-01 0.0e+00 0.0e+00 0.000e+00 p
5.05e-01 -1.8e+01 # -2.9e-15 1.514e-15 f
Best performance (actual): 0.504
</pre>
<p>
The obtained \(\gamma \approx 0.5\) means that it found a controller \(K(s)\) that stabilize the closed-loop system, and such that:
</p>
\begin{aligned}
& \| W_s(s) S(s) \|_\infty < 0.5 \\
& \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
\end{aligned}
<p>
This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure <a href="#org1014010">23</a>.
</p>
<div id="org1014010" class="figure">
<p><img src="figs/results_sensitivity_hinf.png" alt="results_sensitivity_hinf.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Weighting function and obtained closed-loop sensitivity</p>
</div>
</div>
</div>
<div id="outline-container-org5643309" class="outline-3">
<h3 id="org5643309"><span class="section-number-3">4.8</span> Complementary Sensitivity Function</h3>
</div>
<div id="outline-container-orgff0dad8" class="outline-3">
<h3 id="orgff0dad8"><span class="section-number-3">4.9</span> Summary</h3>
<div class="outline-text-3" id="text-4-9">
<table id="org4ba4a2c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</span> Table caption</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">Open-Loop Shaping</th>
<th scope="col" class="org-left">Closed-Loop Shaping</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Reference Tracking</td>
<td class="org-left">\(L\) large</td>
<td class="org-left">\(S\) small</td>
</tr>
<tr>
<td class="org-left">Disturbance Rejection</td>
<td class="org-left">\(L\) large</td>
<td class="org-left">\(SG\) small</td>
</tr>
<tr>
<td class="org-left">Measurement Noise Filtering</td>
<td class="org-left">\(L\) small</td>
<td class="org-left">\(T\) small</td>
</tr>
<tr>
<td class="org-left">Small Command Amplitude</td>
<td class="org-left">\(K\) and \(L\) small</td>
<td class="org-left">\(KS\) small</td>
</tr>
<tr>
<td class="org-left">Robustness</td>
<td class="org-left">Phase/Gain margins</td>
<td class="org-left">Module margin: \(\Vert S\Vert_\infty\) small</td>
</tr>
</tbody>
</table>
<div id="orgfa8f782" class="figure">
<p><img src="figs/h-infinity-4-blocs-constrains.png" alt="h-infinity-4-blocs-constrains.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Shaping the Gang of Four: Limitations</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb9b8856" class="outline-2">
<h2 id="orgb9b8856"><span class="section-number-2">5</span> \(\mathcal{H}_\infty\) Mixed-Sensitivity Synthesis</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org401fcd2"></a>
</p>
</div>
<div id="outline-container-orgaba43d8" class="outline-3">
<h3 id="orgaba43d8"><span class="section-number-3">5.1</span> Problem</h3>
</div>
<div id="outline-container-orgca1e14b" class="outline-3">
<h3 id="orgca1e14b"><span class="section-number-3">5.2</span> Typical Procedure</h3>
</div>
<div id="outline-container-org0fb883d" class="outline-3">
<h3 id="org0fb883d"><span class="section-number-3">5.3</span> Step 1 - Shaping of the Sensitivity Function</h3>
</div>
<div id="outline-container-org4b74053" class="outline-3">
<h3 id="org4b74053"><span class="section-number-3">5.4</span> Step 2 - Shaping of</h3>
</div>
<div id="outline-container-org4acb60b" class="outline-3">
<h3 id="org4acb60b"><span class="section-number-3">5.5</span> General Configuration for various shaping</h3>
<div class="outline-text-3" id="text-5-5">
<details><summary>Shaping of S and KS</summary>
<div id="orgcb7e11c" class="figure">
<p><img src="figs/general_conf_shaping_S_KS.png" alt="general_conf_shaping_S_KS.png" />
</p>
<p><span class="figure-number">Figure 25: </span>Generalized Plant to shape \(S\) and \(KS\)</p>
</div>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 2: </span>General Plant definition corresponding to Figure <a href="#orgcb7e11c">25</a></label><pre class="src src-matlab" id="org4d72ff8">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
0 W2
1 <span class="org-type">-</span>G];
</pre>
</div>
<ul class="org-ul">
<li>\(W_1(s)\) is used to shape \(S\)</li>
<li>\(W_2(s)\) is used to shape \(KS\)</li>
</ul>
</details>
<details><summary>Shaping of S and T</summary>
<div id="org40dfd2e" class="figure">
<p><img src="figs/general_conf_shaping_S_T.png" alt="general_conf_shaping_S_T.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Generalized Plant to shape \(S\) and \(T\)</p>
</div>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 3: </span>General Plant definition corresponding to Figure <a href="#org40dfd2e">26</a></label><pre class="src src-matlab" id="orgf5fe3ae">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
0 G<span class="org-type">*</span>W2
1 <span class="org-type">-</span>G];
</pre>
</div>
<ul class="org-ul">
<li>\(W_1\) is used to shape \(S\)</li>
<li>\(W_2\) is used to shape \(T\)</li>
</ul>
</details>
<details><summary>Shaping of S, T and KS</summary>
<div id="orgcca20dc" class="figure">
<p><img src="figs/general_conf_shaping_S_T_KS.png" alt="general_conf_shaping_S_T_KS.png" />
</p>
<p><span class="figure-number">Figure 27: </span>Generalized Plant to shape \(S\), \(T\) and \(KS\)</p>
</div>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 4: </span>General Plant definition corresponding to Figure <a href="#orgcca20dc">27</a></label><pre class="src src-matlab" id="orgc2d7f73">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
0 W2
0 G<span class="org-type">*</span>W3
1 <span class="org-type">-</span>G];
</pre>
</div>
<ul class="org-ul">
<li>\(W_1\) is used to shape \(S\)</li>
<li>\(W_2\) is used to shape \(KS\)</li>
<li>\(W_3\) is used to shape \(T\)</li>
</ul>
</details>
<details><summary>Shaping of S, T, KS and GS</summary>
<div id="org70a1a89" class="figure">
<p><img src="figs/general_conf_shaping_S_T_KS_GS.png" alt="general_conf_shaping_S_T_KS_GS.png" />
</p>
<p><span class="figure-number">Figure 28: </span>Generalized Plant to shape \(S\), \(T\), \(KS\) and \(GS\)</p>
</div>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 5: </span>General Plant definition corresponding to Figure <a href="#org70a1a89">28</a></label><pre class="src src-matlab" id="org9da7eb9">P = [ W1 <span class="org-type">-</span>W1<span class="org-type">*</span>G<span class="org-type">*</span>W3 <span class="org-type">-</span>G<span class="org-type">*</span>W1
0 0 W2
1 <span class="org-type">-</span>G<span class="org-type">*</span>W3 <span class="org-type">-</span>G];
</pre>
</div>
<ul class="org-ul">
<li>\(W_1\) is used to shape \(S\)</li>
<li>\(W_2\) is used to shape \(KS\)</li>
<li>\(W_1W_3\) is used to shape \(GS\)</li>
<li>\(W_2W_3\) is used to shape \(T\)</li>
</ul>
</details>
</div>
</div>
</div>
<div id="outline-container-org49ed4be" class="outline-2">
<h2 id="org49ed4be"><span class="section-number-2">6</span> Conclusion</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="orgfb5b3fa"></a>
</p>
</div>
</div>
<div id="outline-container-org0386aa4" class="outline-2">
<h2 id="org0386aa4"><span class="section-number-2">7</span> Resources</h2>
<div class="outline-text-2" id="text-7">
<p>
<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>
<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: 2020-11-30 lun. 19:41</p>
</div>
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