<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-03-26 jeu. 17:25 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Control Requirements</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">
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2020 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!--/*--><![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;
}
/*]]>*///-->
</script>
<script>
MathJax = {
tex: { 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">
<a accesskey="h" href="./index.html"> UP </a>
|
<a accesskey="H" href="./index.html"> HOME </a>
</div><div id="content">
<h1 class="title">Control Requirements</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org0341df1">1. Simplify Model for the Nano-Hexapod</a>
<ul>
<li><a href="#org136c9af">1.1. Model of the nano-hexapod</a></li>
<li><a href="#org2fbecfd">1.2. How to include Ground Motion in the model?</a></li>
<li><a href="#org8c1e462">1.3. Motion of the micro-station</a></li>
</ul>
</li>
<li><a href="#org92b1239">2. Control with the Stiff Nano-Hexapod</a>
<ul>
<li><a href="#org19b83b7">2.1. Definition of the values</a></li>
<li><a href="#org0e9811a">2.2. Control using \(d\)</a>
<ul>
<li><a href="#org9828aed">2.2.1. Control Architecture</a></li>
<li><a href="#org5a120e1">2.2.2. Analytical Analysis</a></li>
</ul>
</li>
<li><a href="#orga741e48">2.3. Control using \(F_m\)</a>
<ul>
<li><a href="#org4c74e08">2.3.1. Control Architecture</a></li>
<li><a href="#orgdd5134e">2.3.2. Pure Integrator</a></li>
<li><a href="#org5011ab0">2.3.3. Low pass filter</a></li>
</ul>
</li>
<li><a href="#org4fce174">2.4. Comparison</a></li>
<li><a href="#org5e0585d">2.5. Control using \(x\)</a>
<ul>
<li><a href="#orgfab8395">2.5.1. Analytical analysis</a></li>
<li><a href="#org625e3c2">2.5.2. Control implementation</a></li>
<li><a href="#org3db77f5">2.5.3. Results</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org7c4b4fc">3. Comparison with the use of a Soft nano-hexapod</a></li>
<li><a href="#orgc0253c3">4. Estimate the level of vibration</a></li>
<li><a href="#org764c4a9">5. Requirements on the norm of closed-loop transfer functions</a>
<ul>
<li><a href="#org27379f3">5.1. Approximation of the ASD of perturbations</a></li>
<li><a href="#orgff3d823">5.2. Wanted ASD of outputs</a></li>
<li><a href="#org8c6b37c">5.3. Limiting the bandwidth</a></li>
<li><a href="#org50054f2">5.4. Generalized Weighted plant</a></li>
<li><a href="#org949ab66">5.5. Synthesis</a></li>
<li><a href="#orgfe970e4">5.6. Loop Gain</a></li>
<li><a href="#org5211e8c">5.7. Results</a></li>
<li><a href="#orgb18d7df">5.8. Requirements</a></li>
</ul>
</li>
</ul>
</div>
</div>
<p>
The goal here is to write clear specifications for the NASS.
</p>
<p>
This can then be used for the control synthesis and for the design of the nano-hexapod.
</p>
<p>
Ideal, specifications on the norm of closed loop transfer function should be written.
</p>
<div id="outline-container-org0341df1" class="outline-2">
<h2 id="org0341df1"><span class="section-number-2">1</span> Simplify Model for the Nano-Hexapod</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org136c9af" class="outline-3">
<h3 id="org136c9af"><span class="section-number-3">1.1</span> Model of the nano-hexapod</h3>
<div class="outline-text-3" id="text-1-1">
<p>
Let’s consider the simple mechanical system in Figure <a href="#orgfa3391a">1</a>.
</p>
<div id="orgfa3391a" class="figure">
<p><img src="figs/nass_simple_model.png" alt="nass_simple_model.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Simplified mechanical system for the nano-hexapod</p>
</div>
<p>
The signals are described in table <a href="#orgd89e830">1</a>.
</p>
<table id="orgd89e830" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Signals definition for the generalized plant</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-left"><b>Symbol</b></th>
<th scope="col" class="org-left"><b>Meaning</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><b>Exogenous Inputs</b></td>
<td class="org-left">\(x_\mu\)</td>
<td class="org-left">Motion of the $ν$-hexapod’s base</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">\(F_d\)</td>
<td class="org-left">External Forces applied to the Payload</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">\(r\)</td>
<td class="org-left">Reference signal for tracking</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Exogenous Outputs</b></td>
<td class="org-left">\(x\)</td>
<td class="org-left">Absolute Motion of the Payload</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Sensed Outputs</b></td>
<td class="org-left">\(F_m\)</td>
<td class="org-left">Force Sensors in each leg</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">\(d\)</td>
<td class="org-left">Measured displacement of each leg</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">\(x\)</td>
<td class="org-left">Absolute Motion of the Payload</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Control Signals</b></td>
<td class="org-left">\(F\)</td>
<td class="org-left">Actuator Inputs</td>
</tr>
</tbody>
</table>
<p>
For the nano-hexapod alone, we have the following equations:
\[ \begin{align*}
x &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \\
F_m &= \frac{ms^2}{ms^2 + k} F - \frac{k}{ms^2 + k} F_d + \frac{k m s^2}{ms^2 + k} x_\mu \\
d &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d - \frac{ms^2}{ms^2 + k} x_\mu
\end{align*} \]
</p>
<p>
We can write the equations function of \(\omega_\nu = \sqrt{\frac{k}{m}}\):
\[ \begin{align*}
x &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
F_m &= \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} F - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{k \frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
d &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d - \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu
\end{align*} \]
</p>
<p>
<b>Assumptions</b>:
</p>
<ul class="org-ul">
<li>the forces applied by the nano-hexapod have no influence on the micro-station, specifically on the displacement of the top platform of the micro-hexapod.</li>
</ul>
<p>
This means that the nano-hexapod can be considered separately from the micro-station and that the motion \(x_\mu\) is imposed and considered as an external input.
</p>
<p>
The nano-hexapod can thus be represented as in Figure <a href="#orgb2d1168">2</a>.
</p>
<div id="orgb2d1168" class="figure">
<p><img src="figs/nano_station_inputs_outputs.png" alt="nano_station_inputs_outputs.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Block representation of the nano-hexapod</p>
</div>
</div>
</div>
<div id="outline-container-org2fbecfd" class="outline-3">
<h3 id="org2fbecfd"><span class="section-number-3">1.2</span> How to include Ground Motion in the model?</h3>
<div class="outline-text-3" id="text-1-2">
<p>
What we measure is not the absolute motion \(x\), but the relative motion \(x - w\) where \(w\) is the motion of the granite.
</p>
<p>
Also, \(w\) induces some motion \(x_\mu\) through the transmissibility of the micro-station.
</p>
</div>
</div>
<div id="outline-container-org8c1e462" class="outline-3">
<h3 id="org8c1e462"><span class="section-number-3">1.3</span> Motion of the micro-station</h3>
<div class="outline-text-3" id="text-1-3">
<p>
As explained, we consider \(x_\mu\) as an external input (\(F\) has no influence on \(x_\mu\)).
</p>
<p>
\(x_\mu\) is the motion of the micro-station’s top platform due to the motion of each stage of the micro-station.
</p>
<p>
We consider that \(x_\mu\) has the following form:
\[ x_\mu = T_\mu r + d_\mu \]
where:
</p>
<ul class="org-ul">
<li>\(T_\mu r\) corresponds to the response of the stages due to the reference \(r\)</li>
<li>\(d_\mu\) is the motion of the hexapod due to all the vibrations of the stages</li>
</ul>
<p>
\(T_\mu\) can be considered to be a low pass filter with a bandwidth corresponding approximatively to the bandwidth of the micro-station’s stages.
To simplify, we can consider \(T_\mu\) to be a first order low pass filter:
\[ T_\mu = \frac{1}{1 + s/\omega_\mu} \]
where \(\omega_\mu\) corresponds to the tracking speed of the micro-station.
</p>
<p>
What is important to note is that while \(x_\mu\) is viewed as a perturbation from the nano-hexapod point of view, \(x_\mu\) <b>does</b> depend on the reference signal \(r\).
</p>
<p>
Also, here, we suppose that the granite is not moving.
</p>
<p>
If we now include the motion of the granite \(w\), we obtain the block diagram shown in Figure <a href="#org974c98f">3</a>.
</p>
<div id="org974c98f" class="figure">
<p><img src="figs/nano_station_ground_motion.png" alt="nano_station_ground_motion.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Ground Motion \(w\) included</p>
</div>
<p>
\(T_w\) is the mechanical transmissibility of the micro-station.
We can approximate this transfer function by a second order low pass filter:
\[ T_w = \frac{1}{1 + 2 \xi s/\omega_0 + s^2/\omega_0^2} \]
</p>
</div>
</div>
</div>
<div id="outline-container-org92b1239" class="outline-2">
<h2 id="org92b1239"><span class="section-number-2">2</span> Control with the Stiff Nano-Hexapod</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org19b83b7" class="outline-3">
<h3 id="org19b83b7"><span class="section-number-3">2.1</span> Definition of the values</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Let’s define the mass and stiffness of the nano-hexapod.
</p>
<div class="org-src-container">
<pre class="src src-matlab">m = 50; <span class="org-comment">% [kg]</span>
k = 1e7; <span class="org-comment">% [N/m]</span>
</pre>
</div>
<p>
Let’s define the Plant as shown in Figure <a href="#orgb2d1168">2</a>:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gn = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> k)<span class="org-type">*</span>[<span class="org-type">-</span>k, k<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, m<span class="org-type">*</span>s<span class="org-type">^</span>2; 1, <span class="org-type">-</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, 1; 1, k, 1];
Gn.InputName = {<span class="org-string">'Fd'</span>, <span class="org-string">'xmu'</span>, <span class="org-string">'F'</span>};
Gn.OutputName = {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'x'</span>};
</pre>
</div>
<p>
Now, define the transmissibility transfer function \(T_\mu\) corresponding to the micro-station motion.
</p>
<div class="org-src-container">
<pre class="src src-matlab">wmu = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% [rad/s]</span>
Tmu = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wmu);
Tmu.InputName = {<span class="org-string">'r1'</span>};
Tmu.OutputName = {<span class="org-string">'ymu'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>40;
xi = 0.5;
Tw = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2);
Tw.InputName = {<span class="org-string">'w1'</span>};
Tw.OutputName = {<span class="org-string">'dw'</span>};
</pre>
</div>
<p>
We add the fact that \(x_\mu = d_\mu + T_\mu r + T_w w\):
</p>
<div class="org-src-container">
<pre class="src src-matlab">Wsplit = [tf(1); tf(1)];
Wsplit.InputName = {<span class="org-string">'w'</span>};
Wsplit.OutputName = {<span class="org-string">'w1'</span>, <span class="org-string">'w2'</span>};
S = sumblk(<span class="org-string">'xmu = ymu + dmu + dw'</span>);
Sw = sumblk(<span class="org-string">'y = x - w2'</span>);
Gpz = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {<span class="org-string">'Fd'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'r1'</span>, <span class="org-string">'F'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'y'</span>});
</pre>
</div>
</div>
</div>
<div id="outline-container-org0e9811a" class="outline-3">
<h3 id="org0e9811a"><span class="section-number-3">2.2</span> Control using \(d\)</h3>
<div class="outline-text-3" id="text-2-2">
</div>
<div id="outline-container-org9828aed" class="outline-4">
<h4 id="org9828aed"><span class="section-number-4">2.2.1</span> Control Architecture</h4>
<div class="outline-text-4" id="text-2-2-1">
<p>
Let’s consider a feedback loop using \(d\) as shown in Figure <a href="#orgb50386a">4</a>.
</p>
<div id="orgb50386a" class="figure">
<p><img src="figs/nano_station_control_d.png" alt="nano_station_control_d.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Feedback diagram using \(d\)</p>
</div>
</div>
</div>
<div id="outline-container-org5a120e1" class="outline-4">
<h4 id="org5a120e1"><span class="section-number-4">2.2.2</span> Analytical Analysis</h4>
<div class="outline-text-4" id="text-2-2-2">
<p>
Let’s apply a direct velocity feedback by deriving \(d\):
\[ F = F^\prime - g s d \]
</p>
<p>
Thus:
\[ d = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d - \frac{ms^2}{ms^2 + gs + k} x_\mu \]
</p>
<p>
\[ F = \frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu \]
</p>
<p>
and
\[ x = \frac{1}{ms^2 + k} (\frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu) + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \]
</p>
<p>
\[ x = \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F^\prime + \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F_d + \frac{mgs^3 + k(ms^2 + gs + k)}{(ms^2 + k) (ms^2 + gs + k)} x_\mu \]
</p>
<p>
And we finally obtain:
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} x_\mu \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_dvf = 2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m)<span class="org-type">*</span>s;
K_dvf.InputName = {<span class="org-string">'d'</span>};
K_dvf.OutputName = {<span class="org-string">'F'</span>};
Gpz_dvf = feedback(Gpz, K_dvf, <span class="org-string">'name'</span>);
</pre>
</div>
<p>
Now let’s consider that \(x_\mu = d_\mu + T_\mu r\)
</p>
<p>
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + T_\mu \frac{gs + k}{ms^2 + gs + k} r \]
</p>
<p>
And \(\epsilon = r - x\):
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 + gs + k - T_\mu (gs + k)}{ms^2 + gs + k} r \]
</p>
<div class="important">
<p>
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 - S_\mu(gs + k)}{ms^2 + gs + k} r \]
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orga741e48" class="outline-3">
<h3 id="orga741e48"><span class="section-number-3">2.3</span> Control using \(F_m\)</h3>
<div class="outline-text-3" id="text-2-3">
</div>
<div id="outline-container-org4c74e08" class="outline-4">
<h4 id="org4c74e08"><span class="section-number-4">2.3.1</span> Control Architecture</h4>
<div class="outline-text-4" id="text-2-3-1">
<p>
Let’s consider a feedback loop using \(Fm\) as shown in Figure <a href="#org5012ef2">5</a>.
</p>
<div id="org5012ef2" class="figure">
<p><img src="figs/nano_station_control_Fm.png" alt="nano_station_control_Fm.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Feedback diagram using \(F_m\)</p>
</div>
</div>
</div>
<div id="outline-container-orgdd5134e" class="outline-4">
<h4 id="orgdd5134e"><span class="section-number-4">2.3.2</span> Pure Integrator</h4>
<div class="outline-text-4" id="text-2-3-2">
<p>
Let’s apply integral force feedback by integration \(F_m\):
\[ F = F^\prime - \frac{g}{s} F_m \]
</p>
<p>
And we finally obtain:
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} x_\mu \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = 2<span class="org-type">*</span>sqrt(k<span class="org-type">/</span>m)<span class="org-type">/</span>s;
K_iff.InputName = {<span class="org-string">'Fm'</span>};
K_iff.OutputName = {<span class="org-string">'F'</span>};
Gpz_iff = feedback(Gpz, K_iff, <span class="org-string">'name'</span>);
</pre>
</div>
<p>
Now let’s consider that \(x_\mu = d_\mu + T_\mu r\)
</p>
<p>
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{T_\mu k}{ms^2 + mgs + k} r \]
</p>
<p>
And \(\epsilon = r - x\):
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + k - T_\mu k}{ms^2 + mgs + k} r \]
</p>
<div class="important">
<p>
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + S_\mu k}{ms^2 + mgs + k} r \]
</p>
</div>
</div>
</div>
<div id="outline-container-org5011ab0" class="outline-4">
<h4 id="org5011ab0"><span class="section-number-4">2.3.3</span> Low pass filter</h4>
<div class="outline-text-4" id="text-2-3-3">
<p>
Instead of a pure integrator, let’s use a low pass filter with a cut-off frequency above the bandwidth of the micro-station \(\omega_mu\)
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% K_iff = (2*sqrt(k/m)/(2*wmu))*(1/(1 + s/(2*wmu)));</span>
<span class="org-comment">% K_iff.InputName = {'Fm'};</span>
<span class="org-comment">% K_iff.OutputName = {'F'};</span>
<span class="org-comment">% Gpz_iff = feedback(Gpz, K_iff, 'name');</span>
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org4fce174" class="outline-3">
<h3 id="org4fce174"><span class="section-number-3">2.4</span> Comparison</h3>
<div class="outline-text-3" id="text-2-4">
<div id="orgc10daac" class="figure">
<p><img src="figs/comp_iff_dvf_simplified.png" alt="comp_iff_dvf_simplified.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Obtained transfer functions for DVF and IFF (<a href="./figs/comp_iff_dvf_simplified.png">png</a>, <a href="./figs/comp_iff_dvf_simplified.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<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"> </th>
<th scope="col" class="org-left">\(d_\mu\)</th>
<th scope="col" class="org-left">\(F_d\)</th>
<th scope="col" class="org-left">\(w\)</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">IFF</td>
<td class="org-left">Better filtering of the vibrations</td>
<td class="org-left">More sensitive to External forces</td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">DVF</td>
<td class="org-left">Opposite</td>
<td class="org-left">Opposite</td>
<td class="org-left">Little bit better at low frequencies</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org5e0585d" class="outline-3">
<h3 id="org5e0585d"><span class="section-number-3">2.5</span> Control using \(x\)</h3>
<div class="outline-text-3" id="text-2-5">
</div>
<div id="outline-container-orgfab8395" class="outline-4">
<h4 id="orgfab8395"><span class="section-number-4">2.5.1</span> Analytical analysis</h4>
<div class="outline-text-4" id="text-2-5-1">
<p>
Let’s first consider that only the output \(x\) is used for feedback (Figure <a href="#orgd366408">7</a>)
</p>
<div id="orgd366408" class="figure">
<p><img src="figs/nano_station_control_x.png" alt="nano_station_control_x.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Feedback diagram using \(x\)</p>
</div>
<p>
We then have:
\[ \epsilon &= r - G_{\frac{x}{F}} K \epsilon - G_{\frac{x}{F_d}} F_d - G_{\frac{x}{x_\mu}} d_\mu - G_{\frac{x}{x_\mu}} T_\mu r \]
</p>
<p>
And then:
</p>
<div class="important">
<p>
\[ \epsilon = \frac{-G_{\frac{x}{F_d}}}{1 + G_{\frac{x}{F}}K} F_d + \frac{-G_{\frac{x}{x_\mu}}}{1 + G_{\frac{x}{F}}K} d_\mu + \frac{1 - G_{\frac{x}{x_\mu}} T_\mu}{1 + G_{\frac{x}{F}}K} r \]
</p>
</div>
<p>
With \(S = \frac{1}{1 + G_{\frac{x}{F}} K}\), we have:
\[ \epsilon = - S G_{\frac{x}{F_d}} F_d - S G_{\frac{x}{x_\mu}} d_\mu + S (1 - G_{\frac{x}{x_\mu}} T_\mu) r \]
</p>
<p>
We have 3 terms that we would like to have small by design:
</p>
<ul class="org-ul">
<li>\(G_{\frac{x}{F_d}} = \frac{1}{ms^2 + k}\): thus \(k\) and \(m\) should be high to lower the effect of direct forces \(F_d\)</li>
<li>\(G_{\frac{x}{x_\mu}} = \frac{k}{ms^2 + k} = \frac{1}{1 + \frac{s^2}{\omega_\nu^2}}\): \(\omega_\nu\) should be small enough such that it filters out the vibrations of the micro-station</li>
<li>\(1 - G_{\frac{x}{x_\mu}} T_\mu\)</li>
</ul>
<p>
\[ 1 - G_{\frac{x}{x_\mu}} T_\mu = 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} T_\mu \]
</p>
<p>
We can approximate \(T_\mu \approx \frac{1}{1 + \frac{s}{\omega_\mu}}\) to have:
</p>
\begin{align*}
1 - G_{\frac{x}{x_\mu}} T_\mu &= 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} \frac{1}{1 + \frac{s}{\omega_\mu}} \\
&\approx \frac{\frac{s}{\omega_\mu}}{1 + \frac{s}{\omega_\mu}} = S_\mu \text{ if } \omega_\nu > \omega_\mu \\
&\approx \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} = \text{ if } \omega_\nu < \omega_\mu
\end{align*}
<p>
In our case, we have \(\omega_\nu > \omega_\mu\) and thus we cannot lower this term.
</p>
<p>
Some implications on the design are summarized on table <a href="#orga5207fc">2</a>.
</p>
<table id="orga5207fc" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Design recommendation</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Exogenous Outputs</th>
<th scope="col" class="org-left">Design recommendation</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(F_d\)</td>
<td class="org-left">high \(k\), high \(m\)</td>
</tr>
<tr>
<td class="org-left">\(d_\mu\)</td>
<td class="org-left">low \(k\), high \(m\)</td>
</tr>
<tr>
<td class="org-left">\(r\)</td>
<td class="org-left">no influence if \(\omega_\nu > \omega_\mu\)</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org625e3c2" class="outline-4">
<h4 id="org625e3c2"><span class="section-number-4">2.5.2</span> Control implementation</h4>
<div class="outline-text-4" id="text-2-5-2">
<p>
Controller for the damped plant using DVF.
</p>
<div class="org-src-container">
<pre class="src src-matlab">wb = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% control bandwidth [rad/s]</span>
<span class="org-comment">% Lead</span>
h = 2.0;
wz = wb<span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
wp = wb<span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
H = 1<span class="org-type">/</span>h<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wz)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wp);
<span class="org-comment">% Integrator until 10Hz</span>
Hi = (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>(s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10);
K = Hi<span class="org-type">*</span>H<span class="org-type">*</span>(1<span class="org-type">/</span>s);
Kpz_dvf = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kpz_dvf.InputName = {<span class="org-string">'e'</span>};
Kpz_dvf.OutputName = {<span class="org-string">'Fi'</span>};
</pre>
</div>
<p>
Controller for the damped plant using IFF.
</p>
<div class="org-src-container">
<pre class="src src-matlab">wb = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% control bandwidth [rad/s]</span>
<span class="org-comment">% Lead</span>
h = 2.0;
wz = wb<span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
wp = wb<span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
H = 1<span class="org-type">/</span>h<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wz)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wp);
<span class="org-comment">% Integrator until 10Hz</span>
Hi = (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>(s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10);
K = Hi<span class="org-type">*</span>H<span class="org-type">*</span>(1<span class="org-type">/</span>s);
Kpz_iff = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kpz_iff.InputName = {<span class="org-string">'e'</span>};
Kpz_iff.OutputName = {<span class="org-string">'Fi'</span>};
</pre>
</div>
<p>
Loop gain
</p>
<div id="org0d0fb80" class="figure">
<p><img src="figs/simple_loop_gain_pz.png" alt="simple_loop_gain_pz.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Loop Gain (<a href="./figs/simple_loop_gain_pz.png">png</a>, <a href="./figs/simple_loop_gain_pz.pdf">pdf</a>)</p>
</div>
<p>
Let’s connect all the systems as shown in Figure <a href="#orgd366408">7</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>);
R = [tf(1); tf(1)];
R.InputName = {<span class="org-string">'r'</span>};
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
F = [tf(1); tf(1)];
F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
Gpz_fb_dvf = connect(Gpz_dvf, Kpz_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
Gpz_fb_iff = connect(Gpz_iff, Kpz_iff, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
</pre>
</div>
</div>
</div>
<div id="outline-container-org3db77f5" class="outline-4">
<h4 id="org3db77f5"><span class="section-number-4">2.5.3</span> Results</h4>
<div class="outline-text-4" id="text-2-5-3">
<div id="org2b4e783" class="figure">
<p><img src="figs/simple_hac_lac_results.png" alt="simple_hac_lac_results.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results.png">png</a>, <a href="./figs/simple_hac_lac_results.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<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"> </th>
<th scope="col" class="org-left">Reference Tracking</th>
<th scope="col" class="org-left">Vibration Filtering</th>
<th scope="col" class="org-left">Compliance</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">DVF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left"> </td>
<td class="org-left">Better for \(\omega < \omega_\nu\)</td>
</tr>
<tr>
<td class="org-left">IFF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left">Better for \(\omega > \omega_\nu\)</td>
<td class="org-left"> </td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div id="outline-container-org7c4b4fc" class="outline-2">
<h2 id="org7c4b4fc"><span class="section-number-2">3</span> Comparison with the use of a Soft nano-hexapod</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-matlab">m = 50; <span class="org-comment">% [kg]</span>
k = 1e3; <span class="org-comment">% [N/m]</span>
Gn = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> k)<span class="org-type">*</span>[<span class="org-type">-</span>k, k<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, m<span class="org-type">*</span>s<span class="org-type">^</span>2; 1, <span class="org-type">-</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, 1; 1, k, 1];
Gn.InputName = {<span class="org-string">'Fd'</span>, <span class="org-string">'xmu'</span>, <span class="org-string">'F'</span>};
Gn.OutputName = {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'x'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">wmu = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% [rad/s]</span>
Tmu = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wmu);
Tmu.InputName = {<span class="org-string">'r1'</span>};
Tmu.OutputName = {<span class="org-string">'ymu'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>40;
xi = 0.5;
Tw = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2);
Tw.InputName = {<span class="org-string">'w1'</span>};
Tw.OutputName = {<span class="org-string">'dw'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Wsplit = [tf(1); tf(1)];
Wsplit.InputName = {<span class="org-string">'w'</span>};
Wsplit.OutputName = {<span class="org-string">'w1'</span>, <span class="org-string">'w2'</span>};
S = sumblk(<span class="org-string">'xmu = ymu + dmu + dw'</span>);
Sw = sumblk(<span class="org-string">'y = x - w2'</span>);
Gvc = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {<span class="org-string">'Fd'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'r1'</span>, <span class="org-string">'F'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'y'</span>});
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Kvc_dvf = 2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m)<span class="org-type">*</span>s;
Kvc_dvf.InputName = {<span class="org-string">'d'</span>};
Kvc_dvf.OutputName = {<span class="org-string">'F'</span>};
Gvc_dvf = feedback(Gvc, Kvc_dvf, <span class="org-string">'name'</span>);
Kvc_iff = 2<span class="org-type">*</span>sqrt(k<span class="org-type">/</span>m)<span class="org-type">/</span>s;
Kvc_iff.InputName = {<span class="org-string">'Fm'</span>};
Kvc_iff.OutputName = {<span class="org-string">'F'</span>};
Gvc_iff = feedback(Gvc, Kvc_iff, <span class="org-string">'name'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">wb = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100; <span class="org-comment">% control bandwidth [rad/s]</span>
<span class="org-comment">% Lead</span>
h = 2.0;
wz = wb<span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
wp = wb<span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
H = 1<span class="org-type">/</span>h<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wz)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wp);
Kvc_dvf = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kvc_dvf.InputName = {<span class="org-string">'e'</span>};
Kvc_dvf.OutputName = {<span class="org-string">'Fi'</span>};
Kvc_iff = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kvc_iff.InputName = {<span class="org-string">'e'</span>};
Kvc_iff.OutputName = {<span class="org-string">'Fi'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>);
R = [tf(1); tf(1)];
R.InputName = {<span class="org-string">'r'</span>};
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
F = [tf(1); tf(1)];
F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
Gvc_fb_dvf = connect(Gvc_dvf, Kvc_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
</pre>
</div>
<div id="org3817d8a" class="figure">
<p><img src="figs/simple_hac_lac_results_soft.png" alt="simple_hac_lac_results_soft.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results_soft.png">png</a>, <a href="./figs/simple_hac_lac_results_soft.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<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"> </th>
<th scope="col" class="org-left">Reference Tracking</th>
<th scope="col" class="org-left">Vibration Filtering</th>
<th scope="col" class="org-left">Compliance</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">DVF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left"> </td>
<td class="org-left">Better for \(\omega < \omega_\nu\)</td>
</tr>
<tr>
<td class="org-left">IFF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left">Better for \(\omega > \omega_\nu\)</td>
<td class="org-left"> </td>
</tr>
</tbody>
</table>
<div id="org55e0fe2" class="figure">
<p><img src="figs/simple_comp_vc_pz.png" alt="simple_comp_vc_pz.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Comparison of the closed-loop transfer functions for Soft and Stiff nano-hexapod (<a href="./figs/simple_comp_vc_pz.png">png</a>, <a href="./figs/simple_comp_vc_pz.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-center" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-center"><b>Soft</b></th>
<th scope="col" class="org-center"><b>Stiff</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><b>Reference Tracking</b></td>
<td class="org-center">=</td>
<td class="org-center">=</td>
</tr>
<tr>
<td class="org-left"><b>Ground Motion</b></td>
<td class="org-center">=</td>
<td class="org-center">=</td>
</tr>
<tr>
<td class="org-left"><b>Vibration Isolation</b></td>
<td class="org-center">+</td>
<td class="org-center">-</td>
</tr>
<tr>
<td class="org-left"><b>Compliance</b></td>
<td class="org-center">-</td>
<td class="org-center">+</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgc0253c3" class="outline-2">
<h2 id="orgc0253c3"><span class="section-number-2">4</span> Estimate the level of vibration</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-matlab">gm = load(<span class="org-string">'./mat/psd_gm.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'psd_gm'</span>);
rz = load(<span class="org-string">'./mat/pxsp_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxsp_r'</span>);
tyz = load(<span class="org-string">'./mat/pxz_ty_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxz_ty_r'</span>);
</pre>
</div>
<p>
If we note the PSD \(\Gamma\):
\[ \Gamma_y = |G_{\frac{y}{w}}|^2 \Gamma_w + |G_{\frac{y}{x_\mu}}|^2 \Gamma_{x_\mu} \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">x_pz = abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
x_vc = abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
</pre>
</div>
<div id="org858053c" class="figure">
<p><img src="figs/simple_asd_motion_error.png" alt="simple_asd_motion_error.png" />
</p>
<p><span class="figure-number">Figure 12: </span>ASD of the position error due to Ground Motion and Vibration (<a href="./figs/simple_asd_motion_error.png">png</a>, <a href="./figs/simple_asd_motion_error.pdf">pdf</a>)</p>
</div>
<p>
Actuator usage
</p>
<div class="org-src-container">
<pre class="src src-matlab">F_pz = abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
F_vc = abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">sqrt(trapz(f, F_pz))
sqrt(trapz(f, F_vc))
</pre>
</div>
<pre class="example">
sqrt(trapz(f, F_pz))
ans =
84.8961762069446
sqrt(trapz(f, F_vc))
ans =
0.0387785981815527
</pre>
</div>
</div>
<div id="outline-container-org764c4a9" class="outline-2">
<h2 id="org764c4a9"><span class="section-number-2">5</span> Requirements on the norm of closed-loop transfer functions</h2>
<div class="outline-text-2" id="text-5">
</div>
<div id="outline-container-org27379f3" class="outline-3">
<h3 id="org27379f3"><span class="section-number-3">5.1</span> Approximation of the ASD of perturbations</h3>
<div class="outline-text-3" id="text-5-1">
<div class="org-src-container">
<pre class="src src-matlab">G_rz = 1e<span class="org-type">-</span>9<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>0.5)<span class="org-type">^</span>2<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>(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>100)<span class="org-type">^</span>2));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G_gm = 1e<span class="org-type">-</span>8<span class="org-type">*</span>1<span class="org-type">/</span>s<span class="org-type">^</span>2<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>2<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>10)<span class="org-type">^</span>3));
</pre>
</div>
</div>
</div>
<div id="outline-container-orgff3d823" class="outline-3">
<h3 id="orgff3d823"><span class="section-number-3">5.2</span> Wanted ASD of outputs</h3>
<div class="outline-text-3" id="text-5-2">
<p>
Wanted ASD of motion error
</p>
<div class="org-src-container">
<pre class="src src-matlab">y_wanted = 100e<span class="org-type">-</span>9; <span class="org-comment">% 10nm rms wanted</span>
y_bw = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100; <span class="org-comment">% bandwidth [rad/s]</span>
G_y = 2<span class="org-type">*</span>y_wanted<span class="org-type">/</span>sqrt(y_bw) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>y_bw<span class="org-type">/</span>10) <span class="org-type">/</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>y_bw);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2))
</pre>
</div>
<pre class="example">
sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))
ans =
9.47118350214793e-08
</pre>
</div>
</div>
<div id="outline-container-org8c6b37c" class="outline-3">
<h3 id="org8c6b37c"><span class="section-number-3">5.3</span> Limiting the bandwidth</h3>
<div class="outline-text-3" id="text-5-3">
<div class="org-src-container">
<pre class="src src-matlab">wF = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10;
G_F = 100000<span class="org-type">*</span>(wF <span class="org-type">+</span> s)<span class="org-type">^</span>2;
</pre>
</div>
</div>
</div>
<div id="outline-container-org50054f2" class="outline-3">
<h3 id="org50054f2"><span class="section-number-3">5.4</span> Generalized Weighted plant</h3>
<div class="outline-text-3" id="text-5-4">
<p>
Let’s create a generalized weighted plant for controller synthesis.
</p>
<p>
Let’s start simple:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-left"><b>Symbol</b></th>
<th scope="col" class="org-left"><b>Meaning</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><b>Exogenous Inputs</b></td>
<td class="org-left">\(x_\mu\)</td>
<td class="org-left">Motion of the $ν$-hexapod’s base</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Exogenous Outputs</b></td>
<td class="org-left">\(y\)</td>
<td class="org-left">Motion error of the Payload</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Sensed Outputs</b></td>
<td class="org-left">\(y\)</td>
<td class="org-left">Motion error of the Payload</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Control Signals</b></td>
<td class="org-left">\(F\)</td>
<td class="org-left">Actuator Inputs</td>
</tr>
</tbody>
</table>
<p>
Add \(F\) as output.
</p>
<div class="org-src-container">
<pre class="src src-matlab">F = [tf(1); tf(1)];
F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
P_pz = connect(F, Gpz_dvf, {<span class="org-string">'dmu'</span>, <span class="org-string">'Fi'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'Fu'</span>, <span class="org-string">'y'</span>})
P_vc = connect(F, Gvc_dvf, {<span class="org-string">'dmu'</span>, <span class="org-string">'Fi'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'Fu'</span>, <span class="org-string">'y'</span>})
</pre>
</div>
<p>
Normalization.
</p>
<p>
We multiply the plant input by \(G_{rz}\) and the plant output by \(G_y^{-1}\):
</p>
<div class="org-src-container">
<pre class="src src-matlab">P_pz_norm = blkdiag(inv(G_y), inv(G_F), 1)<span class="org-type">*</span>P_pz<span class="org-type">*</span>blkdiag(G_rz, 1);
P_pz_norm.OutputName = {<span class="org-string">'z'</span>, <span class="org-string">'F'</span>, <span class="org-string">'y'</span>};
P_pz_norm.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'u'</span>};
P_vc_norm = blkdiag(inv(G_y), inv(G_F), 1)<span class="org-type">*</span>P_vc<span class="org-type">*</span>blkdiag(G_rz, 1);
P_vc_norm.OutputName = {<span class="org-string">'z'</span>, <span class="org-string">'F'</span>, <span class="org-string">'y'</span>};
P_vc_norm.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'u'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-org949ab66" class="outline-3">
<h3 id="org949ab66"><span class="section-number-3">5.5</span> Synthesis</h3>
<div class="outline-text-3" id="text-5-5">
<div class="org-src-container">
<pre class="src src-matlab">[Kpz_dvf,CL_vc,<span class="org-type">~</span>] = hinfsyn(minreal(P_pz_norm), 1, 1, <span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'LMI'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
Kpz_dvf.InputName = {<span class="org-string">'e'</span>};
Kpz_dvf.OutputName = {<span class="org-string">'Fi'</span>};
[Kvc_dvf,CL_pz,<span class="org-type">~</span>] = hinfsyn(minreal(P_vc_norm), 1, 1, <span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'LMI'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
Kvc_dvf.InputName = {<span class="org-string">'e'</span>};
Kvc_dvf.OutputName = {<span class="org-string">'Fi'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-orgfe970e4" class="outline-3">
<h3 id="orgfe970e4"><span class="section-number-3">5.6</span> Loop Gain</h3>
<div class="outline-text-3" id="text-5-6">
<div class="org-src-container">
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>);
R = [tf(1); tf(1)];
R.InputName = {<span class="org-string">'r'</span>};
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
F = [tf(1); tf(1)];
F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
Gpz_fb_dvf = connect(Gpz_dvf, <span class="org-type">-</span>Kpz_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
Gvc_fb_dvf = connect(Gvc_dvf, <span class="org-type">-</span>Kvc_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
</pre>
</div>
</div>
</div>
<div id="outline-container-org5211e8c" class="outline-3">
<h3 id="org5211e8c"><span class="section-number-3">5.7</span> Results</h3>
</div>
<div id="outline-container-orgb18d7df" class="outline-3">
<h3 id="orgb18d7df"><span class="section-number-3">5.8</span> Requirements</h3>
<div class="outline-text-3" id="text-5-8">
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left">reference tracking</td>
<td class="org-left">\(\epsilon/r\)</td>
<td class="org-left">-120dB at 1Hz</td>
</tr>
<tr>
<td class="org-left">vibration isolation</td>
<td class="org-left">\(x/x_\mu\)</td>
<td class="org-left">-60dB above 10Hz</td>
</tr>
<tr>
<td class="org-left">compliance</td>
<td class="org-left">\(x/F_d\)</td>
<td class="org-left"> </td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-26 jeu. 17:25</p>
</div>
</body>
</html>