Add custom css
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.figure p{
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text-align: center;
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}
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.figure img{
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max-width:100%;
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display: block;
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margin: auto;
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}
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table {
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margin-left: auto;
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margin-right: auto;
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}
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@ -513,7 +513,7 @@ legend{
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padding:0;
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white-space:normal}
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.fa:before,#content .admonition-title:before,#content h1 .headerlink:before,#content h2 .headerlink:before,#content h3 .headerlink:before,#content h4 .headerlink:before,#content h5 .headerlink:before,#content h6 .headerlink:before,#content dl dt .headerlink:before,.icon:before,.wy-dropdown .caret:before,.wy-inline-validate.wy-inline-validate-success .wy-input-context:before,.wy-inline-validate.wy-inline-validate-danger .wy-input-context:before,.wy-inline-validate.wy-inline-validate-warning .wy-input-context:before,.wy-inline-validate.wy-inline-validate-info .wy-input-context:before,.wy-alert,#content .note,#content .attention,#content .caution,#content .danger,#content .error,#content .summary,#content .hint,#content .important,#content .tip,#content .warning,#content .question,#content .seealso,#content .admonitiontodo,.btn,input[type="text"],input[type="password"],input[type="email"],input[type="url"],input[type="date"],input[type="month"],input[type="time"],input[type="datetime"],input[type="datetime-local"],input[type="week"],input[type="number"],input[type="search"],input[type="tel"],input[type="color"],select,textarea,#table-of-contents li.on a,#table-of-contents li.current>a,.wy-side-nav-search>a,.wy-side-nav-search .wy-dropdown>a,.wy-nav-top a{
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-webkit-font-smoothing:antialiased}
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/*!
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@ -576,7 +576,7 @@ a .fa,a #content .admonition-title,#content a .admonition-title{
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.nav #content .admonition-title,#content .nav .admonition-title,.nav .icon{
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display:inline}
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.wy-alert,#content .note,#content .attention,#content .caution,#content .danger,#content .error,#content .hint,#content .important,#content .tip,#content .warning,#content .seealso,#content .admonitiontodo{
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padding:12px;
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line-height:24px;
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margin-bottom:24px;
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@ -596,32 +596,45 @@ a .fa,a #content .admonition-title,#content a .admonition-title{
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#content .danger,#content .error{
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font-family:FontAwesome;
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#content .note,#content .question,#content .seealso{
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.wy-alert p:last-child,#content .note p:last-child,#content .attention p:last-child,#content .caution p:last-child,#content .danger p:last-child,#content .error p:last-child,#content .hint p:last-child,#content .important p:last-child,#content .tip p:last-child,#content .warning p:last-child,#content .seealso p:last-child,#content .admonitiontodo p:last-child{
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#content .admonition-title.tip,#content .admonition-title.important,#content .admonition-title.hint{
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#content .admonition-title.tip,#content .admonition-title.important,#content .admonition-title.summary,#content .admonition-title.hint{
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line-height: 1;
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background:#1abc9c}
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#content .important,#content .tip,#content .hint{
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#content .important,#content .tip,#content .summary,#content .hint{
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background:#dbfaf4}
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#content .admonition-title.note,#content .admonition-title.seealso{
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line-height: 1;
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background:#6ab0de}
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@ -938,7 +951,7 @@ footer p{
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font-style:italic;
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}
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#content .note .last,#content .attention .last,#content .caution .last,#content .danger .last,#content .error .last,#content .hint .last,#content .important .last,#content .tip .last,#content .warning .last,#content .seealso .last,#content .admonitiontodo .last{
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#content .note .last,#content .attention .last,#content .caution .last,#content .danger .last,#content .error .last,#content .hint .summary,#content .hint .last,#content .important .last,#content .tip .last,#content .warning .last,#content .question .last,#content .seealso .last,#content .admonitiontodo .last{
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#content .admonition-title:before{
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234
index.html
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index.html
@ -3,14 +3,14 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-11-06 ven. 11:59 -->
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<!-- 2020-11-06 ven. 12:04 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>SVD Control</title>
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<meta name="generator" content="Org mode" />
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<meta name="author" content="Dehaeze Thomas" />
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<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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<link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
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<link rel="stylesheet" type="text/css" href="./css/custom.css"/>
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<script type="text/javascript" src="./js/jquery.min.js"></script>
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<script type="text/javascript" src="./js/bootstrap.min.js"></script>
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<script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
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@ -35,56 +35,56 @@
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org9a4d410">1. Gravimeter - Simscape Model</a>
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<li><a href="#org29eb71f">1. Gravimeter - Simscape Model</a>
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<ul>
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<li><a href="#org5411e8d">1.1. Introduction</a></li>
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<li><a href="#orgda2cb96">1.2. Simscape Model - Parameters</a></li>
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<li><a href="#orgf880bca">1.3. System Identification - Without Gravity</a></li>
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<li><a href="#orga8f3792">1.4. System Identification - With Gravity</a></li>
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<li><a href="#orge91b7c5">1.5. Analytical Model</a>
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<li><a href="#org3d08142">1.1. Introduction</a></li>
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<li><a href="#org0e81328">1.2. Simscape Model - Parameters</a></li>
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<li><a href="#orgfb8bd07">1.3. System Identification - Without Gravity</a></li>
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<li><a href="#org73ffaa0">1.4. System Identification - With Gravity</a></li>
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<li><a href="#org0a007cf">1.5. Analytical Model</a>
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<ul>
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<li><a href="#org1843007">1.5.1. Parameters</a></li>
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<li><a href="#orgf341de2">1.5.2. Generation of the State Space Model</a></li>
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<li><a href="#org19b49ae">1.5.3. Comparison with the Simscape Model</a></li>
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<li><a href="#orgd4f737f">1.5.4. Analysis</a></li>
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<li><a href="#org1f3bdc6">1.5.5. Control Section</a></li>
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<li><a href="#orgbe1dd80">1.5.6. Greshgorin radius</a></li>
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<li><a href="#orgf7383e9">1.5.7. Injecting ground motion in the system to have the output</a></li>
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<li><a href="#orgd57f9d7">1.5.1. Parameters</a></li>
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<li><a href="#orgccfbd49">1.5.2. Generation of the State Space Model</a></li>
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<li><a href="#org29be676">1.5.3. Comparison with the Simscape Model</a></li>
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<li><a href="#orgb216c9b">1.5.4. Analysis</a></li>
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<li><a href="#org09d1753">1.5.5. Control Section</a></li>
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<li><a href="#org08382b3">1.5.6. Greshgorin radius</a></li>
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<li><a href="#org9828644">1.5.7. Injecting ground motion in the system to have the output</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org9303df5">2. Gravimeter - Functions</a>
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<li><a href="#org5cf1b81">2. Gravimeter - Functions</a>
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<ul>
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<li><a href="#orgbac772c">2.1. <code>align</code></a></li>
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<li><a href="#org77a1eca">2.2. <code>pzmap_testCL</code></a></li>
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<li><a href="#org865007e">2.1. <code>align</code></a></li>
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<li><a href="#org1ec1be4">2.2. <code>pzmap_testCL</code></a></li>
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</ul>
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</li>
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<li><a href="#org4fe31f1">3. Stewart Platform - Simscape Model</a>
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<li><a href="#orgae25787">3. Stewart Platform - Simscape Model</a>
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<ul>
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<li><a href="#org3a0372c">3.1. Simscape Model - Parameters</a></li>
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<li><a href="#orgb7ea9cc">3.2. Identification of the plant</a></li>
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<li><a href="#org14b6381">3.3. Obtained Dynamics</a></li>
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<li><a href="#orgbb09f73">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
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<li><a href="#orgd6de0da">3.5. Verification of the decoupling using the “Gershgorin Radii”</a></li>
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<li><a href="#orge2bb3a7">3.6. Decoupled Plant</a></li>
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<li><a href="#org2308148">3.7. Diagonal Controller</a></li>
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<li><a href="#orgcd2163a">3.8. Closed-Loop system Performances</a></li>
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<li><a href="#orgddc1d62">3.1. Simscape Model - Parameters</a></li>
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<li><a href="#org4bf5131">3.2. Identification of the plant</a></li>
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<li><a href="#org1d0f1f6">3.3. Obtained Dynamics</a></li>
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<li><a href="#orge008cca">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
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<li><a href="#org8f626c6">3.5. Verification of the decoupling using the “Gershgorin Radii”</a></li>
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<li><a href="#orgcb6b23d">3.6. Decoupled Plant</a></li>
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<li><a href="#org35975ff">3.7. Diagonal Controller</a></li>
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<li><a href="#orgfc10dab">3.8. Closed-Loop system Performances</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org9a4d410" class="outline-2">
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<h2 id="org9a4d410"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
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<div id="outline-container-org29eb71f" class="outline-2">
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<h2 id="org29eb71f"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org5411e8d" class="outline-3">
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<h3 id="org5411e8d"><span class="section-number-3">1.1</span> Introduction</h3>
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<div id="outline-container-org3d08142" class="outline-3">
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<h3 id="org3d08142"><span class="section-number-3">1.1</span> Introduction</h3>
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<div class="outline-text-3" id="text-1-1">
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<div id="org7952bff" class="figure">
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<div id="org405c3e2" class="figure">
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<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
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@ -92,8 +92,8 @@
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</div>
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</div>
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<div id="outline-container-orgda2cb96" class="outline-3">
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<h3 id="orgda2cb96"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
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<div id="outline-container-org0e81328" class="outline-3">
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<h3 id="org0e81328"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>)
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@ -124,8 +124,8 @@ g = 0; <span class="org-comment">% Gravity [m/s2]</span>
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</div>
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</div>
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<div id="outline-container-orgf880bca" class="outline-3">
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<h3 id="orgf880bca"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
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<div id="outline-container-orgfb8bd07" class="outline-3">
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||||
<h3 id="orgfb8bd07"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
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||||
<div class="outline-text-3" id="text-1-3">
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||||
<div class="org-src-container">
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||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
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@ -147,7 +147,7 @@ G.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">
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</pre>
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||||
</div>
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||||
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||||
<pre class="example" id="org8470c34">
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||||
<pre class="example" id="org9b4cec2">
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||||
pole(G)
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ans =
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||||
-0.000473481142385795 + 21.7596190728632i
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||||
@ -172,7 +172,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
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||||
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||||
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||||
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||||
<div id="orge61197a" class="figure">
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||||
<div id="orge565e71" class="figure">
|
||||
<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
|
||||
@ -180,8 +180,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga8f3792" class="outline-3">
|
||||
<h3 id="orga8f3792"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
|
||||
<div id="outline-container-org73ffaa0" class="outline-3">
|
||||
<h3 id="org73ffaa0"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
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||||
<div class="outline-text-3" id="text-1-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span>
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||||
@ -198,7 +198,7 @@ Gg.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string"
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||||
<p>
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||||
We can now see that the system is unstable due to gravity.
|
||||
</p>
|
||||
<pre class="example" id="org75a5ebf">
|
||||
<pre class="example" id="orga648ec2">
|
||||
pole(Gg)
|
||||
ans =
|
||||
-10.9848275341252 + 0i
|
||||
@ -210,7 +210,7 @@ ans =
|
||||
</pre>
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||||
|
||||
|
||||
<div id="org5b5c53a" class="figure">
|
||||
<div id="orgddeeca1" class="figure">
|
||||
<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
|
||||
@ -218,12 +218,12 @@ ans =
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||||
</div>
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||||
</div>
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||||
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||||
<div id="outline-container-orge91b7c5" class="outline-3">
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||||
<h3 id="orge91b7c5"><span class="section-number-3">1.5</span> Analytical Model</h3>
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||||
<div id="outline-container-org0a007cf" class="outline-3">
|
||||
<h3 id="org0a007cf"><span class="section-number-3">1.5</span> Analytical Model</h3>
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||||
<div class="outline-text-3" id="text-1-5">
|
||||
</div>
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||||
<div id="outline-container-org1843007" class="outline-4">
|
||||
<h4 id="org1843007"><span class="section-number-4">1.5.1</span> Parameters</h4>
|
||||
<div id="outline-container-orgd57f9d7" class="outline-4">
|
||||
<h4 id="orgd57f9d7"><span class="section-number-4">1.5.1</span> Parameters</h4>
|
||||
<div class="outline-text-4" id="text-1-5-1">
|
||||
<p>
|
||||
Bode options.
|
||||
@ -255,8 +255,8 @@ Frequency vector.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf341de2" class="outline-4">
|
||||
<h4 id="orgf341de2"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
|
||||
<div id="outline-container-orgccfbd49" class="outline-4">
|
||||
<h4 id="orgccfbd49"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
|
||||
<div class="outline-text-4" id="text-1-5-2">
|
||||
<p>
|
||||
Mass matrix
|
||||
@ -357,11 +357,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org19b49ae" class="outline-4">
|
||||
<h4 id="org19b49ae"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
|
||||
<div id="outline-container-org29be676" class="outline-4">
|
||||
<h4 id="org29be676"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
|
||||
<div class="outline-text-4" id="text-1-5-3">
|
||||
|
||||
<div id="org12c0516" class="figure">
|
||||
<div id="orgd9d1275" class="figure">
|
||||
<p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p>
|
||||
@ -369,8 +369,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd4f737f" class="outline-4">
|
||||
<h4 id="orgd4f737f"><span class="section-number-4">1.5.4</span> Analysis</h4>
|
||||
<div id="outline-container-orgb216c9b" class="outline-4">
|
||||
<h4 id="orgb216c9b"><span class="section-number-4">1.5.4</span> Analysis</h4>
|
||||
<div class="outline-text-4" id="text-1-5-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-comment">% figure</span>
|
||||
@ -438,8 +438,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1f3bdc6" class="outline-4">
|
||||
<h4 id="org1f3bdc6"><span class="section-number-4">1.5.5</span> Control Section</h4>
|
||||
<div id="outline-container-org09d1753" class="outline-4">
|
||||
<h4 id="org09d1753"><span class="section-number-4">1.5.5</span> Control Section</h4>
|
||||
<div class="outline-text-4" id="text-1-5-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10);
|
||||
@ -579,8 +579,8 @@ legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'D
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbe1dd80" class="outline-4">
|
||||
<h4 id="orgbe1dd80"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
|
||||
<div id="outline-container-org08382b3" class="outline-4">
|
||||
<h4 id="org08382b3"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
|
||||
<div class="outline-text-4" id="text-1-5-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w);
|
||||
@ -626,8 +626,8 @@ ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf7383e9" class="outline-4">
|
||||
<h4 id="orgf7383e9"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
|
||||
<div id="outline-container-org9828644" class="outline-4">
|
||||
<h4 id="org9828644"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
|
||||
<div class="outline-text-4" id="text-1-5-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3);
|
||||
@ -683,15 +683,15 @@ rot = PHI(<span class="org-type">:</span>,11,11);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9303df5" class="outline-2">
|
||||
<h2 id="org9303df5"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
|
||||
<div id="outline-container-org5cf1b81" class="outline-2">
|
||||
<h2 id="org5cf1b81"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
</div>
|
||||
<div id="outline-container-orgbac772c" class="outline-3">
|
||||
<h3 id="orgbac772c"><span class="section-number-3">2.1</span> <code>align</code></h3>
|
||||
<div id="outline-container-org865007e" class="outline-3">
|
||||
<h3 id="org865007e"><span class="section-number-3">2.1</span> <code>align</code></h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
<a id="orgd4332c0"></a>
|
||||
<a id="org4775cee"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -720,11 +720,11 @@ This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org77a1eca" class="outline-3">
|
||||
<h3 id="org77a1eca"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
|
||||
<div id="outline-container-org1ec1be4" class="outline-3">
|
||||
<h3 id="org1ec1be4"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
<a id="org27c7e0b"></a>
|
||||
<a id="org3661b72"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -773,15 +773,15 @@ This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4fe31f1" class="outline-2">
|
||||
<h2 id="org4fe31f1"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
|
||||
<div id="outline-container-orgae25787" class="outline-2">
|
||||
<h2 id="orgae25787"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org6ec05bd">5</a>.
|
||||
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org602832a">5</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6ec05bd" class="figure">
|
||||
<div id="org602832a" class="figure">
|
||||
<p><img src="figs/SP_assembly.png" alt="SP_assembly.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Stewart Platform CAD View</p>
|
||||
@ -791,21 +791,21 @@ In this analysis, we wish to applied SVD control to the Stewart Platform shown i
|
||||
The analysis of the SVD control applied to the Stewart platform is performed in the following sections:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Section <a href="#org18c9a86">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
|
||||
<li>Section <a href="#orga8d5029">3.2</a>: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian</li>
|
||||
<li>Section <a href="#org6c8e356">3.3</a>: The identified Dynamics is shown</li>
|
||||
<li>Section <a href="#org5369ce8">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li>
|
||||
<li>Section <a href="#org8aaf502">3.5</a>: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii</li>
|
||||
<li>Section <a href="#org9aecb63">3.6</a>: The dynamics of the decoupled plant is shown</li>
|
||||
<li>Section <a href="#org8f38a00">3.7</a>: A diagonal controller is defined to control the decoupled plant</li>
|
||||
<li>Section <a href="#org0a61882">3.8</a>: Finally, the closed loop system properties are studied</li>
|
||||
<li>Section <a href="#orgac7dad3">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
|
||||
<li>Section <a href="#org9d2dc26">3.2</a>: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian</li>
|
||||
<li>Section <a href="#orgcb0f631">3.3</a>: The identified Dynamics is shown</li>
|
||||
<li>Section <a href="#org60cd341">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li>
|
||||
<li>Section <a href="#org2ab4886">3.5</a>: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii</li>
|
||||
<li>Section <a href="#orgbe91cc3">3.6</a>: The dynamics of the decoupled plant is shown</li>
|
||||
<li>Section <a href="#org71de783">3.7</a>: A diagonal controller is defined to control the decoupled plant</li>
|
||||
<li>Section <a href="#org2d444fe">3.8</a>: Finally, the closed loop system properties are studied</li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="outline-container-org3a0372c" class="outline-3">
|
||||
<h3 id="org3a0372c"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
|
||||
<div id="outline-container-orgddc1d62" class="outline-3">
|
||||
<h3 id="orgddc1d62"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
<a id="org18c9a86"></a>
|
||||
<a id="orgac7dad3"></a>
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>);
|
||||
@ -844,11 +844,11 @@ We load the Jacobian (previously computed from the geometry).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb7ea9cc" class="outline-3">
|
||||
<h3 id="orgb7ea9cc"><span class="section-number-3">3.2</span> Identification of the plant</h3>
|
||||
<div id="outline-container-org4bf5131" class="outline-3">
|
||||
<h3 id="org4bf5131"><span class="section-number-3">3.2</span> Identification of the plant</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
<a id="orga8d5029"></a>
|
||||
<a id="org9d2dc26"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -885,11 +885,11 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
|
||||
|
||||
|
||||
<p>
|
||||
The “centralized” plant \(\bm{G}_x\) is now computed (Figure <a href="#orgdc8eb92">6</a>).
|
||||
The “centralized” plant \(\bm{G}_x\) is now computed (Figure <a href="#org71176d2">6</a>).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgdc8eb92" class="figure">
|
||||
<div id="org71176d2" class="figure">
|
||||
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Centralized control architecture</p>
|
||||
@ -907,22 +907,22 @@ Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string"
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org14b6381" class="outline-3">
|
||||
<h3 id="org14b6381"><span class="section-number-3">3.3</span> Obtained Dynamics</h3>
|
||||
<div id="outline-container-org1d0f1f6" class="outline-3">
|
||||
<h3 id="org1d0f1f6"><span class="section-number-3">3.3</span> Obtained Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
<a id="org6c8e356"></a>
|
||||
<a id="orgcb0f631"></a>
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org9b3956c" class="figure">
|
||||
<div id="orgab61fa1" class="figure">
|
||||
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org2ca723a" class="figure">
|
||||
<div id="orgdeda0eb" class="figure">
|
||||
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
|
||||
@ -930,11 +930,11 @@ Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string"
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbb09f73" class="outline-3">
|
||||
<h3 id="orgbb09f73"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
||||
<div id="outline-container-orge008cca" class="outline-3">
|
||||
<h3 id="orge008cca"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
<a id="org5369ce8"></a>
|
||||
<a id="org60cd341"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1114,11 +1114,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd6de0da" class="outline-3">
|
||||
<h3 id="orgd6de0da"><span class="section-number-3">3.5</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div id="outline-container-org8f626c6" class="outline-3">
|
||||
<h3 id="org8f626c6"><span class="section-number-3">3.5</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
<a id="org8aaf502"></a>
|
||||
<a id="org2ab4886"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1187,7 +1187,7 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org466a903" class="figure">
|
||||
<div id="org8e3dd15" class="figure">
|
||||
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
|
||||
@ -1195,11 +1195,11 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge2bb3a7" class="outline-3">
|
||||
<h3 id="orge2bb3a7"><span class="section-number-3">3.6</span> Decoupled Plant</h3>
|
||||
<div id="outline-container-orgcb6b23d" class="outline-3">
|
||||
<h3 id="orgcb6b23d"><span class="section-number-3">3.6</span> Decoupled Plant</h3>
|
||||
<div class="outline-text-3" id="text-3-6">
|
||||
<p>
|
||||
<a id="org9aecb63"></a>
|
||||
<a id="orgbe91cc3"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1208,14 +1208,14 @@ Let’s see the bode plot of the decoupled plant \(G_d(s)\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org95eefca" class="figure">
|
||||
<div id="org3040990" class="figure">
|
||||
<p><img src="figs/simscape_model_decoupled_plant_svd.png" alt="simscape_model_decoupled_plant_svd.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Decoupled Plant using SVD</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org0b741a8" class="figure">
|
||||
<div id="orgb0513ee" class="figure">
|
||||
<p><img src="figs/simscape_model_decoupled_plant_jacobian.png" alt="simscape_model_decoupled_plant_jacobian.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Decoupled Plant using the Jacobian</p>
|
||||
@ -1223,11 +1223,11 @@ Let’s see the bode plot of the decoupled plant \(G_d(s)\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2308148" class="outline-3">
|
||||
<h3 id="org2308148"><span class="section-number-3">3.7</span> Diagonal Controller</h3>
|
||||
<div id="outline-container-org35975ff" class="outline-3">
|
||||
<h3 id="org35975ff"><span class="section-number-3">3.7</span> Diagonal Controller</h3>
|
||||
<div class="outline-text-3" id="text-3-7">
|
||||
<p>
|
||||
<a id="org8f38a00"></a>
|
||||
<a id="org71de783"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1243,7 +1243,7 @@ K = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<sp
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The control diagram for the centralized control is shown in Figure <a href="#orgdc8eb92">6</a>.
|
||||
The control diagram for the centralized control is shown in Figure <a href="#org71176d2">6</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1252,7 +1252,7 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6306bec" class="figure">
|
||||
<div id="orgfbe6ac1" class="figure">
|
||||
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Control Diagram for the Centralized control</p>
|
||||
@ -1267,11 +1267,11 @@ The feedback system is computed as shown below.
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The SVD control architecture is shown in Figure <a href="#orgd86732f">13</a>.
|
||||
The SVD control architecture is shown in Figure <a href="#org4df0074">13</a>.
|
||||
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
|
||||
</p>
|
||||
|
||||
<div id="orgd86732f" class="figure">
|
||||
<div id="org4df0074" class="figure">
|
||||
<p><img src="figs/svd_control.png" alt="svd_control.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Control Diagram for the SVD control</p>
|
||||
@ -1287,11 +1287,11 @@ The feedback system is computed as shown below.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcd2163a" class="outline-3">
|
||||
<h3 id="orgcd2163a"><span class="section-number-3">3.8</span> Closed-Loop system Performances</h3>
|
||||
<div id="outline-container-orgfc10dab" class="outline-3">
|
||||
<h3 id="orgfc10dab"><span class="section-number-3">3.8</span> Closed-Loop system Performances</h3>
|
||||
<div class="outline-text-3" id="text-3-8">
|
||||
<p>
|
||||
<a id="org0a61882"></a>
|
||||
<a id="org2d444fe"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1322,11 +1322,11 @@ ans =
|
||||
|
||||
|
||||
<p>
|
||||
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#orga86987f">14</a>.
|
||||
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org0481186">14</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga86987f" class="figure">
|
||||
<div id="org0481186" class="figure">
|
||||
<p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Obtained Transmissibility</p>
|
||||
@ -1337,7 +1337,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-11-06 ven. 11:59</p>
|
||||
<p class="date">Created: 2020-11-06 ven. 12:04</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -11,7 +11,7 @@
|
||||
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/custom.css"/>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
|
||||
|
||||
@ -9,6 +9,8 @@ $(function() {
|
||||
$('.hint').before("<p class='admonition-title hint'>Hint</p>");
|
||||
$('.error').before("<p class='admonition-title error'>Error</p>");
|
||||
$('.danger').before("<p class='admonition-title danger'>Danger</p>");
|
||||
$('.question').before("<p class='admonition-title question'>Question</p>");
|
||||
$('.summary').before("<p class='admonition-title hint'>Summary</p>");
|
||||
});
|
||||
|
||||
$( document ).ready(function() {
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|
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Reference in New Issue
Block a user