Add org-setup-file that all org file links to
This commit is contained in:
@@ -4,227 +4,25 @@
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-04-08 mer. 17:22 -->
|
||||
<!-- 2020-04-17 ven. 10:25 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
<title>Control of the NASS with optimal stiffness</title>
|
||||
<meta name="generator" content="Org mode" />
|
||||
<meta name="author" content="Dehaeze Thomas" />
|
||||
<style type="text/css">
|
||||
<!--/*--><![CDATA[/*><!--*/
|
||||
.title { text-align: center;
|
||||
margin-bottom: .2em; }
|
||||
.subtitle { text-align: center;
|
||||
font-size: medium;
|
||||
font-weight: bold;
|
||||
margin-top:0; }
|
||||
.todo { font-family: monospace; color: red; }
|
||||
.done { font-family: monospace; color: green; }
|
||||
.priority { font-family: monospace; color: orange; }
|
||||
.tag { background-color: #eee; font-family: monospace;
|
||||
padding: 2px; font-size: 80%; font-weight: normal; }
|
||||
.timestamp { color: #bebebe; }
|
||||
.timestamp-kwd { color: #5f9ea0; }
|
||||
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
|
||||
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
|
||||
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
|
||||
.underline { text-decoration: underline; }
|
||||
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
|
||||
p.verse { margin-left: 3%; }
|
||||
pre {
|
||||
border: 1px solid #ccc;
|
||||
box-shadow: 3px 3px 3px #eee;
|
||||
padding: 8pt;
|
||||
font-family: monospace;
|
||||
overflow: auto;
|
||||
margin: 1.2em;
|
||||
}
|
||||
pre.src {
|
||||
position: relative;
|
||||
overflow: visible;
|
||||
padding-top: 1.2em;
|
||||
}
|
||||
pre.src:before {
|
||||
display: none;
|
||||
position: absolute;
|
||||
background-color: white;
|
||||
top: -10px;
|
||||
right: 10px;
|
||||
padding: 3px;
|
||||
border: 1px solid black;
|
||||
}
|
||||
pre.src:hover:before { display: inline;}
|
||||
/* Languages per Org manual */
|
||||
pre.src-asymptote:before { content: 'Asymptote'; }
|
||||
pre.src-awk:before { content: 'Awk'; }
|
||||
pre.src-C:before { content: 'C'; }
|
||||
/* pre.src-C++ doesn't work in CSS */
|
||||
pre.src-clojure:before { content: 'Clojure'; }
|
||||
pre.src-css:before { content: 'CSS'; }
|
||||
pre.src-D:before { content: 'D'; }
|
||||
pre.src-ditaa:before { content: 'ditaa'; }
|
||||
pre.src-dot:before { content: 'Graphviz'; }
|
||||
pre.src-calc:before { content: 'Emacs Calc'; }
|
||||
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
|
||||
pre.src-fortran:before { content: 'Fortran'; }
|
||||
pre.src-gnuplot:before { content: 'gnuplot'; }
|
||||
pre.src-haskell:before { content: 'Haskell'; }
|
||||
pre.src-hledger:before { content: 'hledger'; }
|
||||
pre.src-java:before { content: 'Java'; }
|
||||
pre.src-js:before { content: 'Javascript'; }
|
||||
pre.src-latex:before { content: 'LaTeX'; }
|
||||
pre.src-ledger:before { content: 'Ledger'; }
|
||||
pre.src-lisp:before { content: 'Lisp'; }
|
||||
pre.src-lilypond:before { content: 'Lilypond'; }
|
||||
pre.src-lua:before { content: 'Lua'; }
|
||||
pre.src-matlab:before { content: 'MATLAB'; }
|
||||
pre.src-mscgen:before { content: 'Mscgen'; }
|
||||
pre.src-ocaml:before { content: 'Objective Caml'; }
|
||||
pre.src-octave:before { content: 'Octave'; }
|
||||
pre.src-org:before { content: 'Org mode'; }
|
||||
pre.src-oz:before { content: 'OZ'; }
|
||||
pre.src-plantuml:before { content: 'Plantuml'; }
|
||||
pre.src-processing:before { content: 'Processing.js'; }
|
||||
pre.src-python:before { content: 'Python'; }
|
||||
pre.src-R:before { content: 'R'; }
|
||||
pre.src-ruby:before { content: 'Ruby'; }
|
||||
pre.src-sass:before { content: 'Sass'; }
|
||||
pre.src-scheme:before { content: 'Scheme'; }
|
||||
pre.src-screen:before { content: 'Gnu Screen'; }
|
||||
pre.src-sed:before { content: 'Sed'; }
|
||||
pre.src-sh:before { content: 'shell'; }
|
||||
pre.src-sql:before { content: 'SQL'; }
|
||||
pre.src-sqlite:before { content: 'SQLite'; }
|
||||
/* additional languages in org.el's org-babel-load-languages alist */
|
||||
pre.src-forth:before { content: 'Forth'; }
|
||||
pre.src-io:before { content: 'IO'; }
|
||||
pre.src-J:before { content: 'J'; }
|
||||
pre.src-makefile:before { content: 'Makefile'; }
|
||||
pre.src-maxima:before { content: 'Maxima'; }
|
||||
pre.src-perl:before { content: 'Perl'; }
|
||||
pre.src-picolisp:before { content: 'Pico Lisp'; }
|
||||
pre.src-scala:before { content: 'Scala'; }
|
||||
pre.src-shell:before { content: 'Shell Script'; }
|
||||
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
|
||||
/* additional language identifiers per "defun org-babel-execute"
|
||||
in ob-*.el */
|
||||
pre.src-cpp:before { content: 'C++'; }
|
||||
pre.src-abc:before { content: 'ABC'; }
|
||||
pre.src-coq:before { content: 'Coq'; }
|
||||
pre.src-groovy:before { content: 'Groovy'; }
|
||||
/* additional language identifiers from org-babel-shell-names in
|
||||
ob-shell.el: ob-shell is the only babel language using a lambda to put
|
||||
the execution function name together. */
|
||||
pre.src-bash:before { content: 'bash'; }
|
||||
pre.src-csh:before { content: 'csh'; }
|
||||
pre.src-ash:before { content: 'ash'; }
|
||||
pre.src-dash:before { content: 'dash'; }
|
||||
pre.src-ksh:before { content: 'ksh'; }
|
||||
pre.src-mksh:before { content: 'mksh'; }
|
||||
pre.src-posh:before { content: 'posh'; }
|
||||
/* Additional Emacs modes also supported by the LaTeX listings package */
|
||||
pre.src-ada:before { content: 'Ada'; }
|
||||
pre.src-asm:before { content: 'Assembler'; }
|
||||
pre.src-caml:before { content: 'Caml'; }
|
||||
pre.src-delphi:before { content: 'Delphi'; }
|
||||
pre.src-html:before { content: 'HTML'; }
|
||||
pre.src-idl:before { content: 'IDL'; }
|
||||
pre.src-mercury:before { content: 'Mercury'; }
|
||||
pre.src-metapost:before { content: 'MetaPost'; }
|
||||
pre.src-modula-2:before { content: 'Modula-2'; }
|
||||
pre.src-pascal:before { content: 'Pascal'; }
|
||||
pre.src-ps:before { content: 'PostScript'; }
|
||||
pre.src-prolog:before { content: 'Prolog'; }
|
||||
pre.src-simula:before { content: 'Simula'; }
|
||||
pre.src-tcl:before { content: 'tcl'; }
|
||||
pre.src-tex:before { content: 'TeX'; }
|
||||
pre.src-plain-tex:before { content: 'Plain TeX'; }
|
||||
pre.src-verilog:before { content: 'Verilog'; }
|
||||
pre.src-vhdl:before { content: 'VHDL'; }
|
||||
pre.src-xml:before { content: 'XML'; }
|
||||
pre.src-nxml:before { content: 'XML'; }
|
||||
/* add a generic configuration mode; LaTeX export needs an additional
|
||||
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
|
||||
pre.src-conf:before { content: 'Configuration File'; }
|
||||
|
||||
table { border-collapse:collapse; }
|
||||
caption.t-above { caption-side: top; }
|
||||
caption.t-bottom { caption-side: bottom; }
|
||||
td, th { vertical-align:top; }
|
||||
th.org-right { text-align: center; }
|
||||
th.org-left { text-align: center; }
|
||||
th.org-center { text-align: center; }
|
||||
td.org-right { text-align: right; }
|
||||
td.org-left { text-align: left; }
|
||||
td.org-center { text-align: center; }
|
||||
dt { font-weight: bold; }
|
||||
.footpara { display: inline; }
|
||||
.footdef { margin-bottom: 1em; }
|
||||
.figure { padding: 1em; }
|
||||
.figure p { text-align: center; }
|
||||
.equation-container {
|
||||
display: table;
|
||||
text-align: center;
|
||||
width: 100%;
|
||||
}
|
||||
.equation {
|
||||
vertical-align: middle;
|
||||
}
|
||||
.equation-label {
|
||||
display: table-cell;
|
||||
text-align: right;
|
||||
vertical-align: middle;
|
||||
}
|
||||
.inlinetask {
|
||||
padding: 10px;
|
||||
border: 2px solid gray;
|
||||
margin: 10px;
|
||||
background: #ffffcc;
|
||||
}
|
||||
#org-div-home-and-up
|
||||
{ text-align: right; font-size: 70%; white-space: nowrap; }
|
||||
textarea { overflow-x: auto; }
|
||||
.linenr { font-size: smaller }
|
||||
.code-highlighted { background-color: #ffff00; }
|
||||
.org-info-js_info-navigation { border-style: none; }
|
||||
#org-info-js_console-label
|
||||
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
|
||||
.org-info-js_search-highlight
|
||||
{ background-color: #ffff00; color: #000000; font-weight: bold; }
|
||||
.org-svg { width: 90%; }
|
||||
/*]]>*/-->
|
||||
</style>
|
||||
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
||||
<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
||||
<link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
|
||||
<script type="text/javascript" src="./js/jquery.min.js"></script>
|
||||
<script type="text/javascript" src="./js/bootstrap.min.js"></script>
|
||||
<script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
|
||||
<script type="text/javascript" src="./js/readtheorg.js"></script>
|
||||
<script type="text/javascript">
|
||||
// @license magnet:?xt=urn:btih:1f739d935676111cfff4b4693e3816e664797050&dn=gpl-3.0.txt GPL-v3-or-Later
|
||||
<!--/*--><![CDATA[/*><!--*/
|
||||
function CodeHighlightOn(elem, id)
|
||||
{
|
||||
var target = document.getElementById(id);
|
||||
if(null != target) {
|
||||
elem.cacheClassElem = elem.className;
|
||||
elem.cacheClassTarget = target.className;
|
||||
target.className = "code-highlighted";
|
||||
elem.className = "code-highlighted";
|
||||
}
|
||||
}
|
||||
function CodeHighlightOff(elem, id)
|
||||
{
|
||||
var target = document.getElementById(id);
|
||||
if(elem.cacheClassElem)
|
||||
elem.className = elem.cacheClassElem;
|
||||
if(elem.cacheClassTarget)
|
||||
target.className = elem.cacheClassTarget;
|
||||
}
|
||||
/*]]>*///-->
|
||||
// @license-end
|
||||
</script>
|
||||
<script src="./js/jquery.min.js"></script>
|
||||
<script src="./js/bootstrap.min.js"></script>
|
||||
<script src="./js/jquery.stickytableheaders.min.js"></script>
|
||||
<script src="./js/readtheorg.js"></script>
|
||||
<script>MathJax = {
|
||||
tex: {
|
||||
tags: 'ams',
|
||||
macros: {bm: ["\\boldsymbol{#1}",1],}
|
||||
}
|
||||
};
|
||||
</script>
|
||||
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
<div id="org-div-home-and-up">
|
||||
@@ -237,34 +35,58 @@
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#orgb09af41">1. Low Authority Control - Decentralized Integral Force Feedback</a>
|
||||
<li><a href="#org99c5b6d">1. Low Authority Control - Decentralized Direct Velocity Feedback</a>
|
||||
<ul>
|
||||
<li><a href="#orgf3f8aed">1.1. Initialization</a></li>
|
||||
<li><a href="#org93676a4">1.2. Identification</a></li>
|
||||
<li><a href="#org1f2e125">1.3. Controller Design</a></li>
|
||||
<li><a href="#orgc5a1e81">1.2. Identification</a></li>
|
||||
<li><a href="#orgfef1a3f">1.3. Controller Design</a></li>
|
||||
<li><a href="#org3c73014">1.4. Effect of the Low Authority Control on the Primary Plant</a></li>
|
||||
<li><a href="#orgee5dbee">1.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
|
||||
<li><a href="#org27f255e">1.6. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgf825c29">2. Primary Control</a>
|
||||
<li><a href="#org81dc0a8">2. Primary Control in the leg space</a>
|
||||
<ul>
|
||||
<li><a href="#org611a860">2.1. Identification</a></li>
|
||||
<li><a href="#orgfef1a3f">2.2. Controller Design</a></li>
|
||||
<li><a href="#org1e7a412">2.1. Plant in the leg space</a></li>
|
||||
<li><a href="#orgf39520c">2.2. Control in the leg space</a></li>
|
||||
<li><a href="#org16d192f">2.3. Sensibility to Disturbances and Noise Budget</a></li>
|
||||
<li><a href="#org84f68cc">2.4. Simulations</a></li>
|
||||
<li><a href="#orgbeadec8">2.5. Results</a></li>
|
||||
<li><a href="#org87fa1ac">2.6. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org9bd2bf8">3. Primary Control in the task space</a>
|
||||
<ul>
|
||||
<li><a href="#org07b4a9d">3.1. Plant in the task space</a></li>
|
||||
<li><a href="#org7d888f9">3.2. Control in the task space</a>
|
||||
<ul>
|
||||
<li><a href="#orgb28634b">3.2.1. Stability</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org57e2cfd">3.3. Simulation</a></li>
|
||||
<li><a href="#org882e1ac">3.4. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org84f68cc">3. Simulations</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb09af41" class="outline-2">
|
||||
<h2 id="orgb09af41"><span class="section-number-2">1</span> Low Authority Control - Decentralized Integral Force Feedback</h2>
|
||||
<div id="outline-container-org99c5b6d" class="outline-2">
|
||||
<h2 id="org99c5b6d"><span class="section-number-2">1</span> Low Authority Control - Decentralized Direct Velocity Feedback</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
<a id="orgfec42cb"></a>
|
||||
</p>
|
||||
|
||||
<div id="org7f11a74" class="figure">
|
||||
<p><img src="figs/control_architecture_dvf.png" alt="control_architecture_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Low Authority Control: Decentralized Direct Velocity Feedback</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgf3f8aed" class="outline-3">
|
||||
<h3 id="orgf3f8aed"><span class="section-number-3">1.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
|
||||
initializeGranite();
|
||||
@@ -274,43 +96,45 @@ initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
|
||||
initializeSimscapeConfiguration();
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
|
||||
initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We set the references that corresponds to a tomography experiment.
|
||||
We set the stiffness of the payload fixation:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 1);
|
||||
initializeSimscapeConfiguration();
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-iff'</span>);
|
||||
<pre class="src src-matlab">Kp = 1e8; <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org93676a4" class="outline-3">
|
||||
<h3 id="org93676a4"><span class="section-number-3">1.2</span> Identification</h3>
|
||||
<div id="outline-container-orgc5a1e81" class="outline-3">
|
||||
<h3 id="orgc5a1e81"><span class="section-number-3">1.2</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kx = tf(zeros(6));
|
||||
Kiff = tf(zeros(6));
|
||||
<pre class="src src-matlab">K = tf(zeros(6));
|
||||
Kdvf = tf(zeros(6));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We identify the system for the following payload masses:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ms = [1, 10, 50];
|
||||
|
||||
Gm_iff = {zeros(length(Ms), 1)};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The nano-hexapod has the following leg’s stiffness and damping.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'k'</span>, 1e5, <span class="org-string">'c'</span>, 2e2);
|
||||
</pre>
|
||||
@@ -318,45 +142,494 @@ Gm_iff = {zeros(length(Ms), 1)};
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1f2e125" class="outline-3">
|
||||
<h3 id="org1f2e125"><span class="section-number-3">1.3</span> Controller Design</h3>
|
||||
<div id="outline-container-orgfef1a3f" class="outline-3">
|
||||
<h3 id="orgfef1a3f"><span class="section-number-3">1.3</span> Controller Design</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
Root Locus
|
||||
The obtain dynamics from actuators forces \(\tau_i\) to the relative motion of the legs \(d\mathcal{L}_i\) is shown in Figure <a href="#orgdb7af3b">2</a> for the three considered payload masses.
|
||||
</p>
|
||||
|
||||
<div id="orgf657dfe" class="figure">
|
||||
<p><img src="figs/opt_stiff_iff_root_locus.png" alt="opt_stiff_iff_root_locus.png" />
|
||||
<p>
|
||||
The Root Locus is shown in Figure <a href="#org5814b4f">3</a> and wee see that we have unconditional stability.
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Root Locus for the</p>
|
||||
|
||||
<p>
|
||||
In order to choose the gain such that we obtain good damping for all the three payload masses, we plot the damping ration of the modes as a function of the gain for all three payload masses in Figure <a href="#orgb16b2c3">4</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgdb7af3b" class="figure">
|
||||
<p><img src="figs/opt_stiff_dvf_plant.png" alt="opt_stiff_dvf_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Dynamics for the Direct Velocity Feedback active damping for three payload masses</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5814b4f" class="figure">
|
||||
<p><img src="figs/opt_stiff_dvf_root_locus.png" alt="opt_stiff_dvf_root_locus.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Root Locus for the DVF controll for three payload masses</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Damping as function of the gain
|
||||
</p>
|
||||
|
||||
<div id="orgb16b2c3" class="figure">
|
||||
<p><img src="figs/opt_stiff_dvf_damping_gain.png" alt="opt_stiff_dvf_damping_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Damping ratio of the poles as a function of the DVF gain</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kdvf = 5e3<span class="org-type">*</span>s<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1e3)<span class="org-type">*</span>eye(6);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgf825c29" class="outline-2">
|
||||
<h2 id="orgf825c29"><span class="section-number-2">2</span> Primary Control</h2>
|
||||
|
||||
<div id="outline-container-org3c73014" class="outline-3">
|
||||
<h3 id="org3c73014"><span class="section-number-3">1.4</span> Effect of the Low Authority Control on the Primary Plant</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<p>
|
||||
Let’s identify the dynamics from actuator forces \(\bm{\tau}\) to displacement as measured by the metrology \(\bm{\mathcal{X}}\):
|
||||
\[ \bm{G}(s) = \frac{\bm{\mathcal{X}}}{\bm{\tau}} \]
|
||||
We do so both when the DVF is applied and when it is not applied.
|
||||
</p>
|
||||
|
||||
|
||||
<p>
|
||||
Then, we compute the transfer function from forces applied by the actuators \(\bm{\mathcal{F}}\) to the measured position error in the frame of the nano-hexapod \(\bm{\epsilon}_{\mathcal{X}_n}\):
|
||||
\[ \bm{G}_\mathcal{X}(s) = \frac{\bm{\epsilon}_{\mathcal{X}_n}}{\bm{\mathcal{F}}} = \bm{G}(s) \bm{J}^{-T} \]
|
||||
The obtained dynamics is shown in Figure <a href="#org45c1265">5</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<p>
|
||||
And we compute the transfer function from actuator forces \(\bm{\tau}\) to position error of each leg \(\bm{\epsilon}_\mathcal{L}\):
|
||||
\[ \bm{G}_\mathcal{L} = \frac{\bm{\epsilon}_\mathcal{L}}{\bm{\tau}} = \bm{J} \bm{G}(s) \]
|
||||
The obtained dynamics is shown in Figure <a href="#org069e296">6</a>.
|
||||
</p>
|
||||
|
||||
<div id="org45c1265" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_X.png" alt="opt_stiff_primary_plant_damped_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Primary plant in the task space with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org069e296" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_L.png" alt="opt_stiff_primary_plant_damped_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
</div>
|
||||
<p>
|
||||
The coupling (off diagonal elements) of \(\bm{G}_\mathcal{X}\) are shown in Figure <a href="#orgbb4e497">7</a> both when DVF is applied and when it is not.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The coupling does not change a lot with DVF.
|
||||
</p>
|
||||
|
||||
|
||||
<p>
|
||||
The coupling in the space of the legs \(\bm{G}_\mathcal{L}\) are shown in Figure <a href="#orgc43d759">8</a>.
|
||||
The magnitude of the coupling around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgbb4e497" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_coupling_X.png" alt="opt_stiff_primary_plant_damped_coupling_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Coupling in the primary plant in the task with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc43d759" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_coupling_L.png" alt="opt_stiff_primary_plant_damped_coupling_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Coupling in the primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgee5dbee" class="outline-3">
|
||||
<h3 id="orgee5dbee"><span class="section-number-3">1.5</span> Effect of the Low Authority Control on the Sensibility to Disturbances</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<p>
|
||||
We may now see how Decentralized Direct Velocity Feedback changes the sensibility to disturbances, namely:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Ground motion</li>
|
||||
<li>Spindle and Translation stage vibrations</li>
|
||||
<li>Direct forces applied to the sample</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
To simplify the analysis, we here only consider the vertical direction, thus, we will look at the transfer functions:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>from vertical ground motion \(D_{w,z}\) to the vertical position error of the sample \(E_z\)</li>
|
||||
<li>from vertical vibration forces of the spindle \(F_{R_z,z}\) to \(E_z\)</li>
|
||||
<li>from vertical vibration forces of the translation stage \(F_{T_y,z}\) to \(E_z\)</li>
|
||||
<li>from vertical direct forces (such as cable forces) \(F_{d,z}\) to \(E_z\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The norm of these transfer functions are shown in Figure <a href="#org199898b">9</a>.
|
||||
</p>
|
||||
|
||||
<div id="org199898b" class="figure">
|
||||
<p><img src="figs/opt_stiff_sensibility_dist_dvf.png" alt="opt_stiff_sensibility_dist_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org27f255e" class="outline-3">
|
||||
<h3 id="org27f255e"><span class="section-number-3">1.6</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-1-6">
|
||||
<div class="important">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org81dc0a8" class="outline-2">
|
||||
<h2 id="org81dc0a8"><span class="section-number-2">2</span> Primary Control in the leg space</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
</div>
|
||||
<div id="outline-container-org611a860" class="outline-3">
|
||||
<h3 id="org611a860"><span class="section-number-3">2.1</span> Identification</h3>
|
||||
<p>
|
||||
<a id="orgd0beb6a"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section we implement the control architecture shown in Figure <a href="#org7d5c8bc">10</a> consisting of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>an inner loop with a decentralized direct velocity feedback control</li>
|
||||
<li>an outer loop where the controller \(\bm{K}_\mathcal{L}\) is designed in the frame of the legs</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org7d5c8bc" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_dvf_pos_L.png" alt="control_architecture_hac_dvf_pos_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Cascade Control Architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg’s space</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfef1a3f" class="outline-3">
|
||||
<h3 id="orgfef1a3f"><span class="section-number-3">2.2</span> Controller Design</h3>
|
||||
<p>
|
||||
The controller for decentralized direct velocity feedback is the one designed in Section <a href="#orgfec42cb">1</a>.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org1e7a412" class="outline-3">
|
||||
<h3 id="org1e7a412"><span class="section-number-3">2.1</span> Plant in the leg space</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We now loop at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses are shown in Figure <a href="#org23d23ae">11</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The plant dynamics below \(100\ [Hz]\) is only slightly dependent on the payload mass.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org23d23ae" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_L.png" alt="opt_stiff_primary_plant_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org84f68cc" class="outline-2">
|
||||
<h2 id="org84f68cc"><span class="section-number-2">3</span> Simulations</h2>
|
||||
|
||||
<div id="outline-container-orgf39520c" class="outline-3">
|
||||
<h3 id="orgf39520c"><span class="section-number-3">2.2</span> Control in the leg space</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
We design a diagonal controller with all the same diagonal elements.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The requirements for the controller are:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Crossover frequency of around 100Hz</li>
|
||||
<li>Stable for all the considered payload masses</li>
|
||||
<li>Sufficient phase and gain margin</li>
|
||||
<li>Integral action at low frequency</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The design controller is as follows:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Lead centered around the crossover</li>
|
||||
<li>An integrator below 10Hz</li>
|
||||
<li>A low pass filter at 250Hz</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The loop gain is shown in Figure <a href="#orgbcc0acb">12</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">h = 2.0;
|
||||
Kl = 2e7 <span class="org-type">*</span> eye(6) <span class="org-type">*</span> ...
|
||||
1<span class="org-type">/</span>h<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>100<span class="org-type">/</span>h) <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>100<span class="org-type">*</span>h) <span class="org-type">+</span> 1) <span class="org-type">*</span> ...
|
||||
1<span class="org-type">/</span>h<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>200<span class="org-type">/</span>h) <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>200<span class="org-type">*</span>h) <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> 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> ...
|
||||
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>300);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgbcc0acb" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_loop_gain_L.png" alt="opt_stiff_primary_loop_gain_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Loop gain for the primary plant</p>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
|
||||
K = Kl<span class="org-type">*</span>nano_hexapod.J<span class="org-type">*</span>diag([1, 1, 1, 1, 1, 0]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Check the MIMO stability
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org16d192f" class="outline-3">
|
||||
<h3 id="org16d192f"><span class="section-number-3">2.3</span> Sensibility to Disturbances and Noise Budget</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.
|
||||
</p>
|
||||
<p>
|
||||
We compare the norm of these transfer function for the vertical direction when no control is applied and when HAC-LAC control is applied: Figure <a href="#org9650e03">13</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org9650e03" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_senbility_dist.png" alt="opt_stiff_primary_control_L_senbility_dist.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Sensibility to disturbances when the HAC-LAC control is applied</p>
|
||||
</div>
|
||||
<p>
|
||||
Then, we load the Power Spectral Density of the perturbations and we look at the obtained PSD of the displacement error in the vertical direction due to the disturbances:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org32928e0">14</a>: Amplitude Spectral Density of the vertical position error due to both the vertical ground motion and the vertical vibrations of the spindle</li>
|
||||
<li>Figure <a href="#org7fda8f7">15</a>: Comparison of the Amplitude Spectral Density of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
||||
<li>Figure <a href="#org073608b">16</a>: Comparison of the Cumulative Amplitude Spectrum of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org32928e0" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_psd_dist.png" alt="opt_stiff_primary_control_L_psd_dist.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Amplitude Spectral Density of the vertical position error of the sample when the HAC-DVF control is applied due to both the ground motion and spindle vibrations</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7fda8f7" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_psd_tot.png" alt="opt_stiff_primary_control_L_psd_tot.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Amplitude Spectral Density of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org073608b" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_cas_tot.png" alt="opt_stiff_primary_control_L_cas_tot.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Cumulative Amplitude Spectrum of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org84f68cc" class="outline-3">
|
||||
<h3 id="org84f68cc"><span class="section-number-3">2.4</span> Simulations</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
Let’s now simulate a tomography experiment.
|
||||
To do so, we include all disturbances except vibrations of the translation stage.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeDisturbances();
|
||||
initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we run the simulation for all three payload Masses.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgbeadec8" class="outline-3">
|
||||
<h3 id="orgbeadec8"><span class="section-number-3">2.5</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<p>
|
||||
Let’s now see how this controller performs.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
First, we compute the Power Spectral Density of the sample’s position error and we compare it with the open loop case in Figure <a href="#org6cab6ef">17</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Similarly, the Cumulative Amplitude Spectrum is shown in Figure <a href="#org33e9f1a">18</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Finally, the time domain position error signals are shown in Figure <a href="#orgf0f1950">19</a>.
|
||||
</p>
|
||||
|
||||
<div id="org6cab6ef" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_psd_disp_error.png" alt="opt_stiff_hac_dvf_L_psd_disp_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org33e9f1a" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_cas_disp_error.png" alt="opt_stiff_hac_dvf_L_cas_disp_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgf0f1950" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_pos_error.png" alt="opt_stiff_hac_dvf_L_pos_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org87fa1ac" class="outline-3">
|
||||
<h3 id="org87fa1ac"><span class="section-number-3">2.6</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<div class="important">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9bd2bf8" class="outline-2">
|
||||
<h2 id="org9bd2bf8"><span class="section-number-2">3</span> Primary Control in the task space</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="orge9c2f9a"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section, the control architecture shown in Figure <a href="#org7e70ccc">20</a> is applied and consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>an inner Low Authority Control loop consisting of a decentralized direct velocity control controller</li>
|
||||
<li>an outer loop with the primary controller \(\bm{K}_\mathcal{X}\) designed in the task space</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org7e70ccc" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_dvf_pos_X.png" alt="control_architecture_hac_dvf_pos_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>HAC-LAC architecture</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org07b4a9d" class="outline-3">
|
||||
<h3 id="org07b4a9d"><span class="section-number-3">3.1</span> Plant in the task space</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
Let’s look \(\bm{G}_\mathcal{X}(s)\).
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7d888f9" class="outline-3">
|
||||
<h3 id="org7d888f9"><span class="section-number-3">3.2</span> Control in the task space</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kx = tf(zeros(6));
|
||||
|
||||
h = 2.5;
|
||||
Kx<span class="org-type">(1,1) </span>= 3e7 <span class="org-type">*</span> ...
|
||||
1<span class="org-type">/</span>h<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>100<span class="org-type">/</span>h) <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>100<span class="org-type">*</span>h) <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> 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);
|
||||
|
||||
Kx<span class="org-type">(2,2) </span>= Kx(1,1);
|
||||
|
||||
h = 2.5;
|
||||
Kx<span class="org-type">(3,3) </span>= 3e7 <span class="org-type">*</span> ...
|
||||
1<span class="org-type">/</span>h<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>100<span class="org-type">/</span>h) <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>100<span class="org-type">*</span>h) <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> 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);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">h = 1.5;
|
||||
Kx<span class="org-type">(4,4) </span>= 5e5 <span class="org-type">*</span> ...
|
||||
1<span class="org-type">/</span>h<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>100<span class="org-type">/</span>h) <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>100<span class="org-type">*</span>h) <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> 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);
|
||||
|
||||
Kx<span class="org-type">(5,5) </span>= Kx(4,4);
|
||||
|
||||
h = 1.5;
|
||||
Kx<span class="org-type">(6,6) </span>= 5e4 <span class="org-type">*</span> ...
|
||||
1<span class="org-type">/</span>h<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>30<span class="org-type">/</span>h) <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>30<span class="org-type">*</span>h) <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> 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);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb28634b" class="outline-4">
|
||||
<h4 id="orgb28634b"><span class="section-number-4">3.2.1</span> Stability</h4>
|
||||
<div class="outline-text-4" id="text-3-2-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ms)</span>
|
||||
isstable(feedback(Gm_x{<span class="org-constant">i</span>}<span class="org-type">*</span>Kx, eye(6), <span class="org-type">-</span>1))
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org57e2cfd" class="outline-3">
|
||||
<h3 id="org57e2cfd"><span class="section-number-3">3.3</span> Simulation</h3>
|
||||
</div>
|
||||
<div id="outline-container-org882e1ac" class="outline-3">
|
||||
<h3 id="org882e1ac"><span class="section-number-3">3.4</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<div class="important">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-04-08 mer. 17:22</p>
|
||||
<p class="date">Created: 2020-04-17 ven. 10:25</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user