Add license file

LICENSE.txt

@@ -0,0 +1,21 @@
+MIT License
+
+Copyright (c) 2021 Dehaeze Thomas
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.

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index.html

@@ -3,55 +3,77 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-10-03 sam. 19:04 -->
+<!-- 2021-09-10 ven. 11:07 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
-<title>Robust and Optimal Sensor Fusion</title>
-<meta name="generator" content="Org mode" />
+<title>Robust and Optimal Sensor Fusion using Complementary Filters</title>
 <meta name="author" content="Thomas Dehaeze" />
-<link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
-<link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
-<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>
+<meta name="generator" content="Org Mode" />
+<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
+<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
+<style> #content {margin: auto;} </style>
 </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">Robust and Optimal Sensor Fusion
+</div><div id="content" class="content">
+<h1 class="title">Robust and Optimal Sensor Fusion using Complementary Filters
 <br />
 <span class="subtitle">Dehaeze Thomas and Collette Christophe</span>
 </h1>
-
-<div id="outline-container-org0ec4732" class="outline-2">
-<h2 id="org0ec4732">Paper</h2>
-<div class="outline-text-2" id="text-org0ec4732">
+<blockquote>
 <p>
-The PDF version of the paper is accessible <a href="paper/paper.pdf">here</a>.
+<b>Abstract</b>:
+</p>
+</blockquote>
+
+<div id="outline-container-orga601f2f" class="outline-2">
+<h2 id="orga601f2f">Paper (<a href="paper/dehaeze22_optim_robus_compl_filte.pdf">link</a>)</h2>
+<div class="outline-text-2" id="text-orga601f2f">
+<p>
+The paper has been created <a href="https://orgmode.org/">Org Mode</a> (generating <a href="https://www.latex-project.org/">LaTeX</a> code) under <a href="https://www.gnu.org/software/emacs/">Emacs</a>.
 </p>
 </div>
 </div>
 
-<div id="outline-container-orge418504" class="outline-2">
-<h2 id="orge418504">Matlab Scripts</h2>
-<div class="outline-text-2" id="text-orge418504">
+<div id="outline-container-orgd50e329" class="outline-2">
+<h2 id="orgd50e329">Matlab Scripts (<a href="matlab/dehaeze22_optim_robus_compl_filte_matlab.html">link</a>)</h2>
+<div class="outline-text-2" id="text-orgd50e329">
 <p>
-The Matlab scripts that permits to obtain all the results presented in the paper are accessible <a href="matlab/index.html">here</a>.
+All the <a href="https://fr.mathworks.com/">Matlab</a> code that was used for the paper are accessible so that all the results are reproducible.
 </p>
 </div>
 </div>
 
-<div id="outline-container-org99adac0" class="outline-2">
-<h2 id="org99adac0">Tikz Figures</h2>
-<div class="outline-text-2" id="text-org99adac0">
+<div id="outline-container-org3b999bd" class="outline-2">
+<h2 id="org3b999bd">Tikz Figures (<a href="tikz/dehaeze22_optim_robus_compl_filte_tikz.html">link</a>)</h2>
+<div class="outline-text-2" id="text-org3b999bd">
 <p>
-All the figures in the paper are generated using <a href="https://sourceforge.net/projects/pgf/">TikZ</a>. The code snippets that was used to generate the figures are accessible <a href="tikz/index.html">here</a>.
+All the figures for the paper have been generated using <a href="https://sourceforge.net/projects/pgf/">TikZ</a>.
 </p>
 </div>
 </div>
+
+
+<div id="outline-container-org870b921" class="outline-2">
+<h2 id="org870b921">Cite this paper</h2>
+<div class="outline-text-2" id="text-org870b921">
+<p>
+To cite this paper use the following bibtex code.
+</p>
+<div class="org-src-container">
+<pre class="src src-bibtex">
+</pre>
+</div>
+
+<p>
+You can also use the formatted citation below.
+</p>
+<blockquote>
+nil</blockquote>
+</div>
+</div>
 </div>
 </body>
 </html>

index.org

@@ -1,35 +1,50 @@
-#+TITLE: Robust and Optimal Sensor Fusion
+#+TITLE: Robust and Optimal Sensor Fusion using Complementary Filters
 :DRAWER:
 #+SUBTITLE: Dehaeze Thomas and Collette Christophe
 
 #+HTML_LINK_HOME: ../index.html
-#+HTML_LINK_UP: ../index.html
+#+HTML_LINK_UP:   ../index.html
+
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
+#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
+#+HTML_HEAD: <style> #content {margin: auto;} </style>
 
 #+OPTIONS: toc:nil
 #+OPTIONS: html-postamble:nil
-
-#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
-#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
-#+HTML_HEAD: <script src="js/jquery.min.js"></script>
-#+HTML_HEAD: <script src="js/bootstrap.min.js"></script>
-#+HTML_HEAD: <script src="js/jquery.stickytableheaders.min.js"></script>
-#+HTML_HEAD: <script src="js/readtheorg.js"></script>
 :END:
 
-* Paper
+#+begin_quote
+  *Abstract*:
+
+#+end_quote
+
+* Paper ([[file:paper/dehaeze22_optim_robus_compl_filte.pdf][link]])
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
-The PDF version of the paper is accessible [[file:paper/paper.pdf][here]].
+The paper has been created [[https://orgmode.org/][Org Mode]] (generating [[https://www.latex-project.org/][LaTeX]] code) under [[https://www.gnu.org/software/emacs/][Emacs]].
 
-* Matlab Scripts
+* Matlab Scripts ([[file:matlab/dehaeze22_optim_robus_compl_filte_matlab.org][link]])
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
-The Matlab scripts that permits to obtain all the results presented in the paper are accessible [[file:matlab/index.org][here]].
+All the [[https://fr.mathworks.com/][Matlab]] code that was used for the paper are accessible so that all the results are reproducible.
 
-* Tikz Figures
+* Tikz Figures ([[file:tikz/dehaeze22_optim_robus_compl_filte_tikz.org][link]])
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
-All the figures in the paper are generated using [[https://sourceforge.net/projects/pgf/][TikZ]]. The code snippets that was used to generate the figures are accessible [[file:tikz/index.org][here]].
+All the figures for the paper have been generated using [[https://sourceforge.net/projects/pgf/][TikZ]].
+
+
+* Cite this paper
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+To cite this paper use the following bibtex code.
+#+begin_src bibtex
+#+end_src
+
+You can also use the formatted citation below.
+#+begin_quote
+#+end_quote

journal/.latexmkrc

@@ -0,0 +1,98 @@
+#!/bin/env perl
+
+# Shebang is only to get syntax highlighting right across GitLab, GitHub and IDEs.
+# This file is not meant to be run, but read by `latexmk`.
+
+# ======================================================================================
+# Perl `latexmk` configuration file
+# ======================================================================================
+
+# ======================================================================================
+# PDF Generation/Building/Compilation
+# ======================================================================================
+
+@default_files=('dehaeze22_optim_robus_compl_filte.tex');
+
+# PDF-generating modes are:
+# 1: pdflatex, as specified by $pdflatex variable (still largely in use)
+# 2: postscript conversion, as specified by the $ps2pdf variable (useless)
+# 3: dvi conversion, as specified by the $dvipdf variable (useless)
+# 4: lualatex, as specified by the $lualatex variable (best)
+# 5: xelatex, as specified by the $xelatex variable (second best)
+$pdf_mode = 1;
+
+# Treat undefined references and citations as well as multiply defined references as
+# ERRORS instead of WARNINGS.
+# This is only checked in the *last* run, since naturally, there are undefined references
+# in initial runs.
+# This setting is potentially annoying when debugging/editing, but highly desirable
+# in the CI pipeline, where such a warning should result in a failed pipeline, since the
+# final document is incomplete/corrupted.
+#
+# However, I could not eradicate all warnings, so that `latexmk` currently fails with
+# this option enabled.
+# Specifically, `microtype` fails together with `fontawesome`/`fontawesome5`, see:
+# https://tex.stackexchange.com/a/547514/120853
+# The fix in that answer did not help.
+# Setting `verbose=silent` to mute `microtype` warnings did not work.
+# Switching between `fontawesome` and `fontawesome5` did not help.
+$warnings_as_errors = 0;
+
+# Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853
+$show_time = 1;
+
+# Default is 5; we seem to need more owed to the complexity of the document.
+# Actual documents probably don't need this many since they won't use all features,
+# plus won't be compiling from cold each time.
+$max_repeat=7;
+
+# --shell-escape option (execution of code outside of latex) is required for the
+#'svg' package.
+# It converts raw SVG files to the PDF+PDF_TEX combo using InkScape.
+#
+# SyncTeX allows to jump between source (code) and output (PDF) in IDEs with support
+# (many have it). A value of `1` is enabled (gzipped), `-1` is enabled but uncompressed,
+# `0` is off.
+# Testing in VSCode w/ LaTeX Workshop only worked for the compressed version.
+# Adjust this as needed. Of course, only relevant for local use, no effect on a remote
+# CI pipeline (except for slower compilation, probably).
+#
+# %O and %S will forward Options and the Source file, respectively, given to latexmk.
+#
+# `set_tex_cmds` applies to all *latex commands (latex, xelatex, lualatex, ...), so
+# no need to specify these each. This allows to simply change `$pdf_mode` to get a
+# different engine. Check if this works with `latexmk --commands`.
+set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S");
+
+# Use default pdf viewer
+$pdf_previewer = 'zathura';
+
+# option 2 is same as 1 (run biber when necessary), but also deletes the
+# regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original
+# bib file is not available!
+$bibtex_use = 1;  # default: 1
+
+# Change default `biber` call, help catch errors faster/clearer. See
+# https://web.archive.org/web/20200526101657/https://www.semipol.de/2018/06/12/latex-best-practices.html#database-entries
+$bibtex     = "bibtex %S";
+
+# ======================================================================================
+# Auxiliary Files
+# ======================================================================================
+
+# Let latexmk know about generated files, so they can be used to detect if a
+# rerun is required, or be deleted in a cleanup.
+# loe: List of Examples (KOMAScript)
+# lol: List of Listings (`listings` and `minted` packages)
+# run.xml: biber runs
+# glg: glossaries log
+# glstex: generated from glossaries-extra
+push @generated_exts, 'loe', 'lol', 'run.xml', 'glg', 'glstex';
+
+# Also delete the *.glstex files from package glossaries-extra. Problem is,
+# that that package generates files of the form "basename-digit.glstex" if
+# multiple glossaries are present. Latexmk looks for "basename.glstex" and so
+# does not find those. For that purpose, use wildcard.
+# Also delete files generated by gnuplot/pgfplots contour plots
+# (.dat, .script, .table).
+$clean_ext = "%R-*.glstex %R_contourtmp*.*";

journal/dehaeze22_optim_robus_compl_filte.org

@@ -1,25 +1,15 @@
-#+TITLE: Optimal and Robust Sensor Fusion
+#+TITLE: Optimal and Robust Sensor Fusion using Complementary Filters
 :DRAWER:
 #+LATEX_CLASS: IEEEtran
-#+LATEX_CLASS_OPTIONS: [conference]
+#+LATEX_CLASS_OPTIONS: [10pt,final,journal,a4paper]
 #+OPTIONS: toc:nil todo:nil
 #+STARTUP: overview
 
 #+DATE: {{{time(%Y-%m-%d)}}}
 
-#+AUTHOR: @@latex:\IEEEauthorblockN{Dehaeze Thomas}@@
-#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\@@
-#+AUTHOR: @@latex:Grenoble, France\\@@
-#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
-#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
-#+AUTHOR: @@latex:thomas.dehaeze@esrf.fr@@
-#+AUTHOR: @@latex:}\and@@
-#+AUTHOR: @@latex:\IEEEauthorblockN{Collette Christophe}@@
-#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@
-#+AUTHOR: @@latex:\textit{Free University of Brussels}, Belgium\\@@
-#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
-#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
-#+AUTHOR: @@latex:ccollett@ulb.ac.be@@
+#+AUTHOR: @@latex:\author{Thomas~Dehaeze, Mohit~Verma, and~Christophe~Collette@@
+#+AUTHOR: @@latex:\thanks{The authors are with the Precision Mechatronics Laboratory at the University of Liege, Belgium.}@@
+#+AUTHOR: @@latex:\thanks{Corresponding author: thomas.dehaeze@esrf.fr}@@
 #+AUTHOR: @@latex:}@@
 
 #+LATEX_HEADER: \IEEEoverridecommandlockouts
@@ -38,40 +28,44 @@
 #+LATEX_HEADER_EXTRA: \usepackage{showframe}
 
 #+LATEX_HEADER: \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
-
-\bibliographystyle{IEEEtran}
 :END:
 
-* LaTeX Config                                                      :noexport:
-#+begin_src latex :tangle config.tex
-#+end_src
-
 * Build                                                             :noexport:
 #+NAME: startblock
 #+BEGIN_SRC emacs-lisp :results none
-  (add-to-list 'org-latex-classes
-               '("IEEEtran"
-                 "\\documentclass{IEEEtran}"
-                 ("\\section{%s}" . "\\section*{%s}")
-                 ("\\subsection{%s}" . "\\subsection*{%s}")
-                 ("\\subsubsection{%s}" . "\\subsubsection*{%s}")
-                 ("\\paragraph{%s}" . "\\paragraph*{%s}")
-                 ("\\subparagraph{%s}" . "\\subparagraph*{%s}"))
-               )
+(add-to-list 'org-latex-classes
+             '("IEEEtran"
+               "\\documentclass{IEEEtran}"
+               ("\\section{%s}" . "\\section*{%s}")
+               ("\\subsection{%s}" . "\\subsection*{%s}")
+               ("\\subsubsection{%s}" . "\\subsubsection*{%s}")
+               ("\\paragraph{%s}" . "\\paragraph*{%s}")
+               ("\\subparagraph{%s}" . "\\subparagraph*{%s}"))
+             )
 
-  (defun delete-org-comments (backend)
-    (loop for comment in (reverse (org-element-map (org-element-parse-buffer)
-                                      'comment 'identity))
-          do
-          (setf (buffer-substring (org-element-property :begin comment)
-                                  (org-element-property :end comment))
-                "")))
+;; Remove automatic org headings
+(defun my-latex-filter-removeOrgAutoLabels (text backend info)
+  "Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings."
+  (when (org-export-derived-backend-p backend 'latex)
+    (replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text)))
 
-  ;; add to export hook
-  (add-hook 'org-export-before-processing-hook 'delete-org-comments)
+(add-to-list 'org-export-filter-headline-functions
+             'my-latex-filter-removeOrgAutoLabels)
 
-  ;; Remove hypersetup
-  (setq org-latex-with-hyperref nil)
+;; Automatic delete org org-comments
+(defun delete-org-comments (backend)
+  (loop for comment in (reverse (org-element-map (org-element-parse-buffer)
+                                    'comment 'identity))
+        do
+        (setf (buffer-substring (org-element-property :begin comment)
+                                (org-element-property :end comment))
+              "")))
+
+;; add to export hook
+(add-hook 'org-export-before-processing-hook 'delete-org-comments)
+
+;; Remove hypersetup
+(setq org-latex-with-hyperref nil)
 #+END_SRC
 
 * Abstract                                                            :ignore:
@@ -87,6 +81,8 @@
 * Introduction
 <<sec:introduction>>
 
+cite:mahony08_nonlin_compl_filter_special_orthog_group
+
 - Section ref:sec:optimal_fusion
 - Section ref:sec:robust_fusion
 - Section ref:sec:optimal_robust_fusion
@@ -468,7 +464,7 @@ The synthesis objective is to:
 - Minimize the $\mathcal{H}_2$ norm from $w$ to $(z_{2,1}, z_{2,2})$
 
 #+name: fig:mixed_h2_hinf_synthesis
-#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
+#+caption: Generalized plant $P_{\mathcal{H}_2/\mathcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
 #+attr_latex: :scale 1
 [[file:figs/mixed_h2_hinf_synthesis.pdf]]
 
@@ -511,4 +507,10 @@ The synthesis objective is to:
 * Acknowledgment
 
 * Bibliography                                                       :ignore:
+\bibliographystyle{IEEEtran}
 \bibliography{ref}
+
+* Local Variables                                                   :noexport:
+# Local Variables:
+# org-latex-packages-alist: nil
+# End:

journal/dehaeze22_optim_robus_compl_filte.tex

@@ -1,6 +1,6 @@
-% Created 2020-10-05 lun. 15:33
+% Created 2021-09-10 ven. 11:01
 % Intended LaTeX compiler: pdflatex
-\documentclass[conference]{IEEEtran}
+\documentclass[10pt,final,journal,a4paper]{IEEEtran}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{graphicx}
@@ -14,12 +14,6 @@
 \usepackage{amssymb}
 \usepackage{capt-of}
 \usepackage{hyperref}
-\usepackage[most]{tcolorbox}
-\usepackage{bm}
-\usepackage{booktabs}
-\usepackage{tabularx}
-\usepackage{array}
-\usepackage{siunitx}
 \IEEEoverridecommandlockouts
 \usepackage{cite}
 \usepackage{amsmath,amssymb,amsfonts}
@@ -34,9 +28,9 @@
 \renewcommand{\citedash}{--}
 \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
 \usepackage{showframe}
-\author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
-\date{2020-10-05}
-\title{Optimal and Robust Sensor Fusion}
+\author{\author{Thomas~Dehaeze, Mohit~Verma, and~Christophe~Collette \thanks{The authors are with the Precision Mechatronics Laboratory at the University of Liege, Belgium.} \thanks{Corresponding author: thomas.dehaeze@esrf.fr} }}
+\date{2021-09-10}
+\title{Optimal and Robust Sensor Fusion using Complementary Filters}
 \begin{document}
 
 \maketitle
@@ -50,9 +44,10 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
 \end{IEEEkeywords}
 
 \section{Introduction}
-\label{sec:org26a7400}
 \label{sec:introduction}
 
+\cite{mahony08_nonlin_compl_filter_special_orthog_group}
+
 \begin{itemize}
 \item Section \ref{sec:optimal_fusion}
 \item Section \ref{sec:robust_fusion}
@@ -61,11 +56,9 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
 \end{itemize}
 
 \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
-\label{sec:org49e80fd}
 \label{sec:optimal_fusion}
 
 \subsection{Sensor Model}
-\label{sec:org9555932}
 Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
 The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
 
@@ -96,19 +89,18 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensor_model_noise.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensor_model_noise.pdf}
 \caption{\label{fig:sensor_model_noise}Sensor Model}
 \end{figure}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:orga12ae12}
 Let's now consider two sensors measuring the same physical quantity \(x\) but with different dynamics \((G_1, G_2)\) and noise characteristics \((N_1, N_2)\) (Figure \ref{fig:sensor_fusion_noise_arch}).
 
 The noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) are considered to be uncorrelated.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensor_fusion_noise_arch.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensor_fusion_noise_arch.pdf}
 \caption{\label{fig:sensor_fusion_noise_arch}Sensor Fusion Architecture with sensor noise}
 \end{figure}
 
@@ -138,7 +130,6 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
 \end{equation}
 
 \subsection{Super Sensor Noise}
-\label{sec:org924b750}
 Let's note \(n\) the super sensor noise.
 \begin{equation}
   n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
@@ -152,7 +143,6 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
 It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
 
 \subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
-\label{sec:org042a601}
 The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the noise \(n\) of the estimation \(\hat{x}\).
 
 And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
@@ -191,67 +181,63 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/h_two_optimal_fusion.pdf}
+\includegraphics[scale=1,scale=1]{figs/h_two_optimal_fusion.pdf}
 \caption{\label{fig:h_two_optimal_fusion}Generalized plant \(P_{\mathcal{H}_2}\) used for the \(\mathcal{H}_2\) synthesis of complementary filters}
 \end{figure}
 
 \subsection{Example}
-\label{sec:org98c54c2}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensors_nominal_dynamics.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensors_nominal_dynamics.pdf}
 \caption{\label{fig:sensors_nominal_dynamics}Sensor nominal dynamics from the velocity of the object to the output voltage}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensors_noise.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensors_noise.pdf}
 \caption{\label{fig:sensors_noise}Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)}
 \end{figure}
 
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/htwo_comp_filters.pdf}
+\includegraphics[scale=1,scale=1]{figs/htwo_comp_filters.pdf}
 \caption{\label{fig:htwo_comp_filters}Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis}
 \end{figure}
 
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/psd_sensors_htwo_synthesis.pdf}
+\includegraphics[scale=1,scale=1]{figs/psd_sensors_htwo_synthesis.pdf}
 \caption{\label{fig:psd_sensors_htwo_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal}
 \end{figure}
 
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/super_sensor_time_domain_h2.pdf}
+\includegraphics[scale=1,scale=1]{figs/super_sensor_time_domain_h2.pdf}
 \caption{\label{fig:super_sensor_time_domain_h2}Noise of individual sensors and noise of the super sensor}
 \end{figure}
 
 \subsection{Robustness Problem}
-\label{sec:org81a0772}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensors_nominal_dynamics_and_uncertainty.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensors_nominal_dynamics_and_uncertainty.pdf}
 \caption{\label{fig:sensors_nominal_dynamics_and_uncertainty}Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_H2.pdf}
+\includegraphics[scale=1,scale=1]{figs/super_sensor_dynamical_uncertainty_H2.pdf}
 \caption{\label{fig:super_sensor_dynamical_uncertainty_H2}Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis}
 \end{figure}
 
 \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
-\label{sec:org78ced60}
 \label{sec:robust_fusion}
 
 \subsection{Representation of Sensor Dynamical Uncertainty}
-\label{sec:org9df3b01}
 
 In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
 In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
@@ -266,12 +252,11 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensor_model_uncertainty.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensor_model_uncertainty.pdf}
 \caption{\label{fig:sensor_model_uncertainty}Sensor Model including Dynamical Uncertainty}
 \end{figure}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:orgf4531ff}
 Let's consider the sensor fusion architecture shown in Figure \ref{fig:sensor_fusion_arch_uncertainty} where the dynamical uncertainties of both sensors are included.
 
 The super sensor estimate is then:
@@ -291,12 +276,11 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensor_fusion_arch_uncertainty.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensor_fusion_arch_uncertainty.pdf}
 \caption{\label{fig:sensor_fusion_arch_uncertainty}Sensor Fusion Architecture with sensor model uncertainty}
 \end{figure}
 
 \subsection{Super Sensor Dynamical Uncertainty}
-\label{sec:orgf5bb33e}
 The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure \ref{fig:uncertainty_set_super_sensor}.
 
 
@@ -304,14 +288,13 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/uncertainty_set_super_sensor.pdf}
+\includegraphics[scale=1,scale=1]{figs/uncertainty_set_super_sensor.pdf}
 \caption{\label{fig:uncertainty_set_super_sensor}Super Sensor model uncertainty displayed in the complex plane}
 \end{figure}
 
 At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
 
 \subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
-\label{sec:orgf07efa7}
 In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
 
 The goal is to design two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the super sensor noise uncertainty is kept reasonably small.
@@ -352,51 +335,48 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/h_infinity_robust_fusion.pdf}
+\includegraphics[scale=1,scale=1]{figs/h_infinity_robust_fusion.pdf}
 \caption{\label{fig:h_infinity_robust_fusion}Generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) synthesis of complementary filters}
 \end{figure}
 
 \subsection{Example}
-\label{sec:org0ca6ef9}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensors_uncertainty_weights.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensors_uncertainty_weights.pdf}
 \caption{\label{fig:sensors_uncertainty_weights}Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)}
 \end{figure}
 
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/weight_uncertainty_bounds_Wu.pdf}
+\includegraphics[scale=1,scale=1]{figs/weight_uncertainty_bounds_Wu.pdf}
 \caption{\label{fig:weight_uncertainty_bounds_Wu}Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/hinf_comp_filters.pdf}
+\includegraphics[scale=1,scale=1]{figs/hinf_comp_filters.pdf}
 \caption{\label{fig:hinf_comp_filters}Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Hinf.pdf}
+\includegraphics[scale=1,scale=1]{figs/super_sensor_dynamical_uncertainty_Hinf.pdf}
 \caption{\label{fig:super_sensor_dynamical_uncertainty_Hinf}Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/psd_sensors_hinf_synthesis.pdf}
+\includegraphics[scale=1,scale=1]{figs/psd_sensors_hinf_synthesis.pdf}
 \caption{\label{fig:psd_sensors_hinf_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the \(\mathcal{H}_\infty\) synthesis}
 \end{figure}
 
 
 \section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:orgf642e73}
 \label{sec:optimal_robust_fusion}
 
 \subsection{Sensor with noise and model uncertainty}
-\label{sec:org8949812}
 We wish now to combine the two previous synthesis, that is to say
 
 The sensors are now modelled by a white noise with unitary PSD \(\tilde{n}_i\) shaped by a LTI transfer function \(N_i(s)\).
@@ -412,12 +392,11 @@ Multiplying by the inverse of the nominal model of the sensor dynamics gives an
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensor_model_noise_uncertainty.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensor_model_noise_uncertainty.pdf}
 \caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
 \end{figure}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:orgcbc3d54}
 
 For reason of space, the blocks \(\hat{G}_i\) and \(\hat{G}_i^{-1}\) are omitted.
 
@@ -439,12 +418,11 @@ The estimate \(\hat{x}\) of \(x\)
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensor_fusion_arch_full.pdf}
+\includegraphics[scale=1,scale=1]{figs/sensor_fusion_arch_full.pdf}
 \caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty}
 \end{figure}
 
 \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:org9d3f160}
 
 The synthesis objective is to generate two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the uncertainty associated with the super sensor is kept reasonably small and such that the RMS value of super sensors noise is minimized.
 
@@ -474,59 +452,52 @@ The synthesis objective is to:
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/mixed_h2_hinf_synthesis.pdf}
-\caption{\label{fig:mixed_h2_hinf_synthesis}Generalized plant \(P_{\mathcal{H}_2/\matlcal{H}_\infty}\) used for the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of complementary filters}
+\includegraphics[scale=1,scale=1]{figs/mixed_h2_hinf_synthesis.pdf}
+\caption{\label{fig:mixed_h2_hinf_synthesis}Generalized plant \(P_{\mathcal{H}_2/\mathcal{H}_\infty}\) used for the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of complementary filters}
 \end{figure}
 
 \subsection{Example}
-\label{sec:org85f304b}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/htwo_hinf_comp_filters.pdf}
+\includegraphics[scale=1,scale=1]{figs/htwo_hinf_comp_filters.pdf}
 \caption{\label{fig:htwo_hinf_comp_filters}Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/psd_sensors_htwo_hinf_synthesis.pdf}
+\includegraphics[scale=1,scale=1]{figs/psd_sensors_htwo_hinf_synthesis.pdf}
 \caption{\label{fig:psd_sensors_htwo_hinf_synthesis}Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/super_sensor_time_domain_h2_hinf.pdf}
+\includegraphics[scale=1,scale=1]{figs/super_sensor_time_domain_h2_hinf.pdf}
 \caption{\label{fig:super_sensor_time_domain_h2_hinf}Noise of individual sensors and noise of the super sensor}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf}
+\includegraphics[scale=1,scale=1]{figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf}
 \caption{\label{fig:super_sensor_dynamical_uncertainty_Htwo_Hinf}Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
 \end{figure}
 
 \section{Experimental Validation}
-\label{sec:org49bf34a}
 \label{sec:experimental_validation}
 
 \subsection{Experimental Setup}
-\label{sec:orgdd8fce6}
 
 \subsection{Sensor Noise and Dynamical Uncertainty}
-\label{sec:org21add72}
 
 \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:org30521a3}
 
 \subsection{Super Sensor Noise and Dynamical Uncertainty}
-\label{sec:org86cde79}
 
 \section{Conclusion}
-\label{sec:org16245b7}
 \label{sec:conclusion}
 
 \section{Acknowledgment}
-\label{sec:orgd992049}
 
+\bibliographystyle{IEEEtran}
 \bibliography{ref}
 \end{document}

js/jquery.stickytableheaders.min.js

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js/readtheorg.js

@@ -1,85 +0,0 @@
-$(function() {
-    $('.note').before("<p class='admonition-title note'>Note</p>");
-    $('.seealso').before("<p class='admonition-title seealso'>See also</p>");
-    $('.warning').before("<p class='admonition-title warning'>Warning</p>");
-    $('.caution').before("<p class='admonition-title caution'>Caution</p>");
-    $('.attention').before("<p class='admonition-title attention'>Attention</p>");
-    $('.tip').before("<p class='admonition-title tip'>Tip</p>");
-    $('.important').before("<p class='admonition-title important'>Important</p>");
-    $('.hint').before("<p class='admonition-title hint'>Hint</p>");
-    $('.error').before("<p class='admonition-title error'>Error</p>");
-    $('.danger').before("<p class='admonition-title danger'>Danger</p>");
-});
-
-$( document ).ready(function() {
-
-    // Shift nav in mobile when clicking the menu.
-    $(document).on('click', "[data-toggle='wy-nav-top']", function() {
-      $("[data-toggle='wy-nav-shift']").toggleClass("shift");
-      $("[data-toggle='rst-versions']").toggleClass("shift");
-    });
-    // Close menu when you click a link.
-    $(document).on('click', ".wy-menu-vertical .current ul li a", function() {
-      $("[data-toggle='wy-nav-shift']").removeClass("shift");
-      $("[data-toggle='rst-versions']").toggleClass("shift");
-    });
-    $(document).on('click', "[data-toggle='rst-current-version']", function() {
-      $("[data-toggle='rst-versions']").toggleClass("shift-up");
-    });
-    // Make tables responsive
-    $("table.docutils:not(.field-list)").wrap("<div class='wy-table-responsive'></div>");
-});
-
-$( document ).ready(function() {
-    $('#text-table-of-contents ul').first().addClass('nav');
-                                        // ScrollSpy also requires that we use
-                                        // a Bootstrap nav component.
-    $('body').scrollspy({target: '#text-table-of-contents'});
-
-    // add sticky table headers
-    $('table').stickyTableHeaders();
-
-    // set the height of tableOfContents
-    var $postamble = $('#postamble');
-    var $tableOfContents = $('#table-of-contents');
-    $tableOfContents.css({paddingBottom: $postamble.outerHeight()});
-
-    // add TOC button
-    var toggleSidebar = $('<div id="toggle-sidebar"><a href="#table-of-contents"><h2>Table of Contents</h2></a></div>');
-    $('#content').prepend(toggleSidebar);
-
-    // add close button when sidebar showed in mobile screen
-    var closeBtn = $('<a class="close-sidebar" href="#">Close</a>');
-    var tocTitle = $('#table-of-contents').find('h2');
-    tocTitle.append(closeBtn);
-});
-
-window.SphinxRtdTheme = (function (jquery) {
-    var stickyNav = (function () {
-        var navBar,
-            win,
-            stickyNavCssClass = 'stickynav',
-            applyStickNav = function () {
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-                    navBar.addClass(stickyNavCssClass);
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-    return {
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-}($));

matlab/dehaeze22_optim_robus_compl_filte_matlab.org

@@ -1,19 +1,14 @@
 #+TITLE: Robust and Optimal Sensor Fusion - Matlab Computation
 :DRAWER:
-#+HTML_LINK_HOME: ./index.html
-#+HTML_LINK_UP: ./index.html
+#+HTML_LINK_HOME: ../index.html
+#+HTML_LINK_UP:   ../index.html
+
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
+#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
 
 #+BIND: org-latex-image-default-option "scale=1"
 #+BIND: org-latex-image-default-width ""
 
-#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
-#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
-#+HTML_HEAD: <script src="../js/jquery.min.js"></script>
-#+HTML_HEAD: <script src="../js/bootstrap.min.js"></script>
-#+HTML_HEAD: <script src="../js/jquery.stickytableheaders.min.js"></script>
-#+HTML_HEAD: <script src="../js/readtheorg.js"></script>
-#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://cdn.rawgit.com/dreampulse/computer-modern-web-font/master/fonts.css">
-
 #+STARTUP: overview
 #+OPTIONS: toc:2
 
@@ -145,8 +140,10 @@ The true sensor dynamics has some uncertainty associated to it and described in
 Both sensor dynamics in $[\frac{V}{m/s}]$ are shown in Figure [[fig:sensors_nominal_dynamics]].
 #+begin_src matlab :exports none
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
   % Magnitude
-  ax1 = subplot(2,1,1);
+  ax1 = nexttile;
   hold on;
   plot(freqs, abs(squeeze(freqresp(G1, freqs, 'Hz'))), '-', 'DisplayName', '$G_1(j\omega)$');
   plot(freqs, abs(squeeze(freqresp(G2, freqs, 'Hz'))), '-', 'DisplayName', '$G_2(j\omega)$');
@@ -156,7 +153,7 @@ Both sensor dynamics in $[\frac{V}{m/s}]$ are shown in Figure [[fig:sensors_nomi
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plot(freqs, 180/pi*angle(squeeze(freqresp(G1, freqs, 'Hz'))), '-');
   plot(freqs, 180/pi*angle(squeeze(freqresp(G2, freqs, 'Hz'))), '-');
@@ -165,6 +162,7 @@ Both sensor dynamics in $[\frac{V}{m/s}]$ are shown in Figure [[fig:sensors_nomi
   ylim([-180 180]);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
+
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
 #+end_src
@@ -190,7 +188,7 @@ The true sensor dynamics $G_i(s)$ is then described by eqref:eq:sensor_dynamics_
 
 The weights $W_i(s)$ representing the dynamical uncertainty are defined below and their magnitude is shown in Figure [[fig:sensors_uncertainty_weights]].
 #+begin_src matlab
-  W1 = createWeight('n', 2, 'w0', 2*pi*3,   'G0', 2, 'G1', 0.1,     'Gc', 1) * ...
+  W1 = createWeight('n', 2, 'w0', 2*pi*3,   'G0', 2, 'G1', 0.1,   'Gc', 1) * ...
        createWeight('n', 2, 'w0', 2*pi*1e3, 'G0', 1, 'G1', 4/0.1, 'Gc', 1/0.1);
 
   W2 = createWeight('n', 2, 'w0', 2*pi*1e2, 'G0', 0.05, 'G1', 4, 'Gc', 1);
@@ -222,8 +220,10 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
 
 #+begin_src matlab :exports none
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
   % Magnitude
-  ax1 = subplot(2,1,1);
+  ax1 = nexttile;
   hold on;
   plotMagUncertainty(W1, freqs, 'G', G1, 'color_i', 1, 'DisplayName', '$G_1$');
   plotMagUncertainty(W2, freqs, 'G', G2, 'color_i', 2, 'DisplayName', '$G_2$');
@@ -240,7 +240,7 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
   ylim([1e-2, 1e4])
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'G', G1, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'G', G2, 'color_i', 2);
@@ -253,6 +253,7 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
   ylim([-180 180]);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
+
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
 #+end_src
@@ -473,9 +474,10 @@ Finally, $H_1(s)$ is defined as follows
 The obtained complementary filters are shown in Figure [[fig:htwo_comp_filters]].
 #+begin_src matlab :exports none
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
 
   % Magnitude
-  ax1 = subplot(2,1,1);
+  ax1 = nexttile;
   hold on;
   plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), 'DisplayName', '$H_1$');
   plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), 'DisplayName', '$H_2$');
@@ -486,7 +488,7 @@ The obtained complementary filters are shown in Figure [[fig:htwo_comp_filters]]
   legend('location', 'northeast', 'FontSize', 8);
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plot(freqs, 180/pi*angle(squeeze(freqresp(H1, freqs, 'Hz'))));
   plot(freqs, 180/pi*angle(squeeze(freqresp(H2, freqs, 'Hz'))));
@@ -640,8 +642,10 @@ As a result the super sensor signal can not be used for feedback applications ab
   Dphi_ss(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))) > 1) = 360;
 
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
   % Magnitude
-  ax1 = subplot(2,1,1);
+  ax1 = nexttile;
   hold on;
   plotMagUncertainty(W1, freqs, 'color_i', 1, 'DisplayName', '$1 + W_1 \Delta_1$');
   plotMagUncertainty(W2, freqs, 'color_i', 2, 'DisplayName', '$1 + W_2 \Delta_2$');
@@ -657,7 +661,7 @@ As a result the super sensor signal can not be used for feedback applications ab
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -668,6 +672,7 @@ As a result the super sensor signal can not be used for feedback applications ab
   ylim([-180 180]);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
+
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
 #+end_src
@@ -770,8 +775,10 @@ The uncertainty bounds of the two individual sensor as well as the wanted maximu
   Dphi_Wu(abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))) > 1) = 360;
 
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
   % Magnitude
-  ax1 = subplot(2,1,1);
+  ax1 = nexttile;
   hold on;
   plotMagUncertainty(W1, freqs, 'color_i', 1, 'DisplayName', '$1 + W_1 \Delta_1$');
   plotMagUncertainty(W2, freqs, 'color_i', 2, 'DisplayName', '$1 + W_2 \Delta_2$');
@@ -787,7 +794,7 @@ The uncertainty bounds of the two individual sensor as well as the wanted maximu
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -879,29 +886,31 @@ The obtained complementary filters as well as the wanted upper bounds are shown
 
 #+begin_src matlab :exports none
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
 
-  ax1 = subplot(2,1,1);
+  % Magnitude
+  ax1 = nexttile;
   hold on;
   plot(freqs, 1./abs(squeeze(freqresp(Wu*W1, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_1|$');
   plot(freqs, 1./abs(squeeze(freqresp(Wu*W2, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_2|$');
   set(gca,'ColorOrderIndex',1)
   plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$|H_1|$');
   plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$|H_2|$');
-
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Magnitude');
   set(gca, 'XTickLabel',[]);
   legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
 
-  ax2 = subplot(2,1,2);
+  % Phase
+  ax2 = nexttile;
   hold on;
   plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
   plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
   hold off;
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   set(gca, 'XScale', 'log');
-  yticks([-360:90:360]);
+  ylim([-90, 90]); yticks([-360:90:360]);
 
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
@@ -930,8 +939,10 @@ The $\mathcal{H}_\infty$ synthesis thus allows to design filters such that the s
   Dphi_ss(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))) > 1) = 360;
 
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
   % Magnitude
-  ax1 = subplot(2,1,1);
+  ax1 = nexttile;
   hold on;
   plotMagUncertainty(W1, freqs, 'color_i', 1, 'DisplayName', '$1 + W_1 \Delta_1$');
   plotMagUncertainty(W2, freqs, 'color_i', 2, 'DisplayName', '$1 + W_2 \Delta_2$');
@@ -951,7 +962,7 @@ The $\mathcal{H}_\infty$ synthesis thus allows to design filters such that the s
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -964,6 +975,7 @@ The $\mathcal{H}_\infty$ synthesis thus allows to design filters such that the s
   ylim([-180 180]);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
+
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
 #+end_src
@@ -1126,31 +1138,31 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt
 
 #+begin_src matlab :exports none
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
 
-  ax1 = subplot(2,1,1);
+  % Magnitude
+  ax1 = nexttile;
   hold on;
   plot(freqs, 1./abs(squeeze(freqresp(Wu*W1, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_1|$');
   plot(freqs, 1./abs(squeeze(freqresp(Wu*W2, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_2|$');
-
   set(gca,'ColorOrderIndex',1)
   plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
   plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
-
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Magnitude');
-  set(gca, 'XTickLabel',[]);
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
   ylim([1e-3, 2]);
   legend('location', 'southeast', 'FontSize', 8);
 
-  ax2 = subplot(2,1,2);
+  % Phase
+  ax2 = nexttile;
   hold on;
   plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
   plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
   hold off;
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   set(gca, 'XScale', 'log');
-  yticks([-360:90:360]);
+  ylim([-90, 90]); yticks([-360:90:360]);
 
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
@@ -1283,8 +1295,10 @@ The uncertainty on the super sensor's dynamics is shown in Figure [[fig:super_se
   Dphi_ss(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))) > 1) = 360;
 
   figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
   % Magnitude
-  ax1 = subplot(2,1,1);
+  ax1 = nexttile;
   hold on;
   plotMagUncertainty(W1, freqs, 'color_i', 1, 'DisplayName', '$1 + W_1 \Delta_1$');
   plotMagUncertainty(W2, freqs, 'color_i', 2, 'DisplayName', '$1 + W_2 \Delta_2$');
@@ -1304,7 +1318,7 @@ The uncertainty on the super sensor's dynamics is shown in Figure [[fig:super_se
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -1313,10 +1327,10 @@ The uncertainty on the super sensor's dynamics is shown in Figure [[fig:super_se
   plot(freqs,  Dphi_Wu, 'k--');
   plot(freqs, -Dphi_Wu, 'k--');
   set(gca,'xscale','log');
-  yticks(-180:90:180);
-  ylim([-180 180]);
+  ylim([-180 180]); yticks(-180:90:180);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
+
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
 #+end_src

matlab/index.html

@@ -3,18 +3,13 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-10-05 lun. 15:08 -->
+<!-- 2020-11-12 jeu. 10:42 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <title>Robust and Optimal Sensor Fusion - Matlab Computation</title>
 <meta name="generator" content="Org mode" />
 <meta name="author" content="Thomas Dehaeze" />
-<link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
-<link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
-<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>
-<link rel="stylesheet" type="text/css" href="https://cdn.rawgit.com/dreampulse/computer-modern-web-font/master/fonts.css">
+<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
+<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
 <script>MathJax = {
           tex: {
             tags: 'ams',
@@ -26,59 +21,59 @@
 </head>
 <body>
 <div id="org-div-home-and-up">
- <a accesskey="h" href="./index.html"> UP </a>
+ <a accesskey="h" href="../index.html"> UP </a>
  |
- <a accesskey="H" href="./index.html"> HOME </a>
+ <a accesskey="H" href="../index.html"> HOME </a>
 </div><div id="content">
 <h1 class="title">Robust and Optimal Sensor Fusion - Matlab Computation</h1>
 <div id="table-of-contents">
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#org0a492b7">1. Sensor Description</a>
+<li><a href="#orgb558a2f">1. Sensor Description</a>
 <ul>
-<li><a href="#org9575585">1.1. Sensor Dynamics</a></li>
-<li><a href="#orgec8c81d">1.2. Sensor Model Uncertainty</a></li>
-<li><a href="#org81d9a34">1.3. Sensor Noise</a></li>
-<li><a href="#org4e23f6c">1.4. Save Model</a></li>
+<li><a href="#org2e756d8">1.1. Sensor Dynamics</a></li>
+<li><a href="#orgf266a4b">1.2. Sensor Model Uncertainty</a></li>
+<li><a href="#orgb768f02">1.3. Sensor Noise</a></li>
+<li><a href="#org426c21e">1.4. Save Model</a></li>
 </ul>
 </li>
-<li><a href="#org2cab1a2">2. Introduction to Sensor Fusion</a>
+<li><a href="#orgce0ac83">2. Introduction to Sensor Fusion</a>
 <ul>
-<li><a href="#org0cbc92d">2.1. Sensor Fusion Architecture</a></li>
-<li><a href="#orge9e0bd4">2.2. Super Sensor Noise</a></li>
-<li><a href="#orgefb4347">2.3. Super Sensor Dynamical Uncertainty</a></li>
+<li><a href="#orgbb36114">2.1. Sensor Fusion Architecture</a></li>
+<li><a href="#org3a4f4b8">2.2. Super Sensor Noise</a></li>
+<li><a href="#org3f0594e">2.3. Super Sensor Dynamical Uncertainty</a></li>
 </ul>
 </li>
-<li><a href="#org5896b60">3. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</a>
+<li><a href="#org096192f">3. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</a>
 <ul>
-<li><a href="#orgf5a8a84">3.1. \(\mathcal{H}_2\) Synthesis</a></li>
-<li><a href="#org72159df">3.2. Super Sensor Noise</a></li>
-<li><a href="#org8ffba19">3.3. Discrepancy between sensor dynamics and model</a></li>
+<li><a href="#org9188eae">3.1. \(\mathcal{H}_2\) Synthesis</a></li>
+<li><a href="#org6d6d909">3.2. Super Sensor Noise</a></li>
+<li><a href="#org850d99e">3.3. Discrepancy between sensor dynamics and model</a></li>
 </ul>
 </li>
-<li><a href="#org26ea7b1">4. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</a>
+<li><a href="#org6d1270a">4. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</a>
 <ul>
-<li><a href="#org8dfd9d2">4.1. Weighting Function used to bound the super sensor uncertainty</a></li>
-<li><a href="#org7422ade">4.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
-<li><a href="#orga0267c6">4.3. Super sensor uncertainty</a></li>
-<li><a href="#org979fede">4.4. Super sensor noise</a></li>
-<li><a href="#orgda33992">4.5. Conclusion</a></li>
+<li><a href="#orgc5a0221">4.1. Weighting Function used to bound the super sensor uncertainty</a></li>
+<li><a href="#org334f826">4.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
+<li><a href="#org211e432">4.3. Super sensor uncertainty</a></li>
+<li><a href="#org7672e0e">4.4. Super sensor noise</a></li>
+<li><a href="#orgdcb47cf">4.5. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org15afe90">5. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</a>
+<li><a href="#org49a7cc2">5. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</a>
 <ul>
-<li><a href="#org0f81a91">5.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</a></li>
-<li><a href="#org417aabd">5.2. Obtained Super Sensor&rsquo;s noise</a></li>
-<li><a href="#org2dce888">5.3. Obtained Super Sensor&rsquo;s Uncertainty</a></li>
-<li><a href="#org47da78c">5.4. Conclusion</a></li>
+<li><a href="#org7ed8ed7">5.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</a></li>
+<li><a href="#orgfb839cf">5.2. Obtained Super Sensor&rsquo;s noise</a></li>
+<li><a href="#orgb8596ac">5.3. Obtained Super Sensor&rsquo;s Uncertainty</a></li>
+<li><a href="#orgfcfa8a2">5.4. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org0afe5ef">6. Matlab Functions</a>
+<li><a href="#orgf267700">6. Matlab Functions</a>
 <ul>
-<li><a href="#orge81e522">6.1. <code>createWeight</code></a></li>
-<li><a href="#org37ec2b4">6.2. <code>plotMagUncertainty</code></a></li>
-<li><a href="#org9f73572">6.3. <code>plotPhaseUncertainty</code></a></li>
+<li><a href="#org005db82">6.1. <code>createWeight</code></a></li>
+<li><a href="#org816b32a">6.2. <code>plotMagUncertainty</code></a></li>
+<li><a href="#orgddb3773">6.3. <code>plotPhaseUncertainty</code></a></li>
 </ul>
 </li>
 </ul>
@@ -89,27 +84,27 @@
 This document is arranged as follows:
 </p>
 <ul class="org-ul">
-<li>Section <a href="#orgee25d07">1</a>: the sensors are described (dynamics, uncertainty, noise)</li>
-<li>Section <a href="#orga64daad">2</a>: the sensor fusion architecture is described and the super sensor noise and dynamical uncertainty are derived</li>
-<li>Section <a href="#orgdd6b9ce">3</a>: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor&rsquo;s noise is minimized</li>
-<li>Section <a href="#org5d93f37">4</a>: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor&rsquo;s uncertainty is bonded to acceptable values</li>
-<li>Section <a href="#org9f98c16">5</a>: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor&rsquo;s uncertainty and to lower the RMS value of the super sensor&rsquo;s noise</li>
-<li>Section <a href="#orgf41dc8d">6</a>: Matlab functions used for the analysis are described</li>
+<li>Section <a href="#org3cd5225">1</a>: the sensors are described (dynamics, uncertainty, noise)</li>
+<li>Section <a href="#orge87db28">2</a>: the sensor fusion architecture is described and the super sensor noise and dynamical uncertainty are derived</li>
+<li>Section <a href="#orgbaf748a">3</a>: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor&rsquo;s noise is minimized</li>
+<li>Section <a href="#orgea66128">4</a>: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor&rsquo;s uncertainty is bonded to acceptable values</li>
+<li>Section <a href="#org2aeafb3">5</a>: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor&rsquo;s uncertainty and to lower the RMS value of the super sensor&rsquo;s noise</li>
+<li>Section <a href="#org50a28ec">6</a>: Matlab functions used for the analysis are described</li>
 </ul>
 
-<div id="outline-container-org0a492b7" class="outline-2">
-<h2 id="org0a492b7"><span class="section-number-2">1</span> Sensor Description</h2>
+<div id="outline-container-orgb558a2f" class="outline-2">
+<h2 id="orgb558a2f"><span class="section-number-2">1</span> Sensor Description</h2>
 <div class="outline-text-2" id="text-1">
 <p>
-<a id="orgee25d07"></a>
+<a id="org3cd5225"></a>
 </p>
 <p>
-In Figure <a href="#org35e2340">1</a> is shown a schematic of a sensor model that is used in the following study.
+In Figure <a href="#orgd79f98f">1</a> is shown a schematic of a sensor model that is used in the following study.
 In this example, the measured quantity \(x\) is the velocity of an object.
 </p>
 
-<table id="org2fe6194" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
-<caption class="t-above"><span class="table-number">Table 1:</span> Description of signals in Figure <a href="#org35e2340">1</a></caption>
+<table id="org5aafbb9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<caption class="t-above"><span class="table-number">Table 1:</span> Description of signals in Figure <a href="#orgd79f98f">1</a></caption>
 
 <colgroup>
 <col  class="org-left" />
@@ -176,8 +171,8 @@ In this example, the measured quantity \(x\) is the velocity of an object.
 </tbody>
 </table>
 
-<table id="org281ecb3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
-<caption class="t-above"><span class="table-number">Table 2:</span> Description of Systems in Figure <a href="#org35e2340">1</a></caption>
+<table id="org3780b2b" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<caption class="t-above"><span class="table-number">Table 2:</span> Description of Systems in Figure <a href="#orgd79f98f">1</a></caption>
 
 <colgroup>
 <col  class="org-left" />
@@ -221,18 +216,18 @@ In this example, the measured quantity \(x\) is the velocity of an object.
 </table>
 
 
-<div id="org35e2340" class="figure">
+<div id="orgd79f98f" class="figure">
 <p><img src="figs-paper/sensor_model_noise_uncertainty.png" alt="sensor_model_noise_uncertainty.png" />
 </p>
 <p><span class="figure-number">Figure 1: </span>Sensor Model</p>
 </div>
 </div>
 
-<div id="outline-container-org9575585" class="outline-3">
-<h3 id="org9575585"><span class="section-number-3">1.1</span> Sensor Dynamics</h3>
+<div id="outline-container-org2e756d8" class="outline-3">
+<h3 id="org2e756d8"><span class="section-number-3">1.1</span> Sensor Dynamics</h3>
 <div class="outline-text-3" id="text-1-1">
 <p>
-<a id="org52415aa"></a>
+<a id="org01c4832"></a>
 Let&rsquo;s consider two sensors measuring the velocity of an object.
 </p>
 
@@ -263,14 +258,14 @@ G2 = g_pos<span class="org-type">/</span>s<span class="org-type">/</span>(1 <spa
 
 <p>
 These nominal dynamics are also taken as the model of the sensor dynamics.
-The true sensor dynamics has some uncertainty associated to it and described in section <a href="#orgf04439d">1.2</a>.
+The true sensor dynamics has some uncertainty associated to it and described in section <a href="#org3943238">1.2</a>.
 </p>
 
 <p>
-Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org2d1ef0b">2</a>.
+Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org355b3fc">2</a>.
 </p>
 
-<div id="org2d1ef0b" class="figure">
+<div id="org355b3fc" class="figure">
 <p><img src="figs/sensors_nominal_dynamics.png" alt="sensors_nominal_dynamics.png" />
 </p>
 <p><span class="figure-number">Figure 2: </span>Sensor nominal dynamics from the velocity of the object to the output voltage</p>
@@ -278,12 +273,12 @@ Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org2d1
 </div>
 </div>
 
-<div id="outline-container-orgec8c81d" class="outline-3">
-<h3 id="orgec8c81d"><span class="section-number-3">1.2</span> Sensor Model Uncertainty</h3>
+<div id="outline-container-orgf266a4b" class="outline-3">
+<h3 id="orgf266a4b"><span class="section-number-3">1.2</span> Sensor Model Uncertainty</h3>
 <div class="outline-text-3" id="text-1-2">
 <p>
-<a id="orgf04439d"></a>
-The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure <a href="#org35e2340">1</a>).
+<a id="org3943238"></a>
+The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure <a href="#orgd79f98f">1</a>).
 </p>
 
 <p>
@@ -295,10 +290,10 @@ The true sensor dynamics \(G_i(s)\) is then described by \eqref{eq:sensor_dynami
 \end{equation}
 
 <p>
-The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure <a href="#org557e062">3</a>.
+The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure <a href="#org2530a6e">3</a>.
 </p>
 <div class="org-src-container">
-<pre class="src src-matlab">W1 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-string">'w0'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>3,   <span class="org-string">'G0'</span>, 2, <span class="org-string">'G1'</span>, 0.1,     <span class="org-string">'Gc'</span>, 1) <span class="org-type">*</span> ...
+<pre class="src src-matlab">W1 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-string">'w0'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>3,   <span class="org-string">'G0'</span>, 2, <span class="org-string">'G1'</span>, 0.1,   <span class="org-string">'Gc'</span>, 1) <span class="org-type">*</span> ...
      createWeight(<span class="org-string">'n'</span>, 2, <span class="org-string">'w0'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1e3, <span class="org-string">'G0'</span>, 1, <span class="org-string">'G1'</span>, 4<span class="org-type">/</span>0.1, <span class="org-string">'Gc'</span>, 1<span class="org-type">/</span>0.1);
 
 W2 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-string">'w0'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1e2, <span class="org-string">'G0'</span>, 0.05, <span class="org-string">'G1'</span>, 4, <span class="org-string">'Gc'</span>, 1);
@@ -306,18 +301,18 @@ W2 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
 </div>
 
 <p>
-The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure <a href="#orgc218675">4</a>.
+The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure <a href="#org415740c">4</a>.
 </p>
 
 
-<div id="org557e062" class="figure">
+<div id="org2530a6e" class="figure">
 <p><img src="figs/sensors_uncertainty_weights.png" alt="sensors_uncertainty_weights.png" />
 </p>
 <p><span class="figure-number">Figure 3: </span>Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)</p>
 </div>
 
 
-<div id="orgc218675" class="figure">
+<div id="org415740c" class="figure">
 <p><img src="figs/sensors_nominal_dynamics_and_uncertainty.png" alt="sensors_nominal_dynamics_and_uncertainty.png" />
 </p>
 <p><span class="figure-number">Figure 4: </span>Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)</p>
@@ -325,12 +320,12 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
 </div>
 </div>
 
-<div id="outline-container-org81d9a34" class="outline-3">
-<h3 id="org81d9a34"><span class="section-number-3">1.3</span> Sensor Noise</h3>
+<div id="outline-container-orgb768f02" class="outline-3">
+<h3 id="orgb768f02"><span class="section-number-3">1.3</span> Sensor Noise</h3>
 <div class="outline-text-3" id="text-1-3">
 <p>
-<a id="org71b587e"></a>
-The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure <a href="#org35e2340">1</a>).
+<a id="orgfd557c8"></a>
+The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure <a href="#orgd79f98f">1</a>).
 </p>
 \begin{equation}
   \Phi_{\tilde{n}_i}(\omega) = 1 \label{eq:unitary_noise_psd}
@@ -344,7 +339,7 @@ The Power Spectral Density of the sensor noise \(\Phi_{n_i}(\omega)\) is then co
 \end{equation}
 
 <p>
-The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure <a href="#org51f9788">5</a>.
+The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure <a href="#org5e2cd13">5</a>.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">omegac = 0.15<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>; G0 = 1e<span class="org-type">-</span>1; Ginf = 1e<span class="org-type">-</span>6;
@@ -356,7 +351,7 @@ N2 = (Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac
 </div>
 
 
-<div id="org51f9788" class="figure">
+<div id="org5e2cd13" class="figure">
 <p><img src="figs/sensors_noise.png" alt="sensors_noise.png" />
 </p>
 <p><span class="figure-number">Figure 5: </span>Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)</p>
@@ -364,8 +359,8 @@ N2 = (Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac
 </div>
 </div>
 
-<div id="outline-container-org4e23f6c" class="outline-3">
-<h3 id="org4e23f6c"><span class="section-number-3">1.4</span> Save Model</h3>
+<div id="outline-container-org426c21e" class="outline-3">
+<h3 id="org426c21e"><span class="section-number-3">1.4</span> Save Model</h3>
 <div class="outline-text-3" id="text-1-4">
 <p>
 All the dynamical systems representing the sensors are saved for further use.
@@ -379,27 +374,27 @@ All the dynamical systems representing the sensors are saved for further use.
 </div>
 </div>
 
-<div id="outline-container-org2cab1a2" class="outline-2">
-<h2 id="org2cab1a2"><span class="section-number-2">2</span> Introduction to Sensor Fusion</h2>
+<div id="outline-container-orgce0ac83" class="outline-2">
+<h2 id="orgce0ac83"><span class="section-number-2">2</span> Introduction to Sensor Fusion</h2>
 <div class="outline-text-2" id="text-2">
 <p>
-<a id="orga64daad"></a>
+<a id="orge87db28"></a>
 </p>
 </div>
 
-<div id="outline-container-org0cbc92d" class="outline-3">
-<h3 id="org0cbc92d"><span class="section-number-3">2.1</span> Sensor Fusion Architecture</h3>
+<div id="outline-container-orgbb36114" class="outline-3">
+<h3 id="orgbb36114"><span class="section-number-3">2.1</span> Sensor Fusion Architecture</h3>
 <div class="outline-text-3" id="text-2-1">
 <p>
-<a id="org31e00a0"></a>
+<a id="org80d2a51"></a>
 </p>
 
 <p>
-The two sensors presented in Section <a href="#orgee25d07">1</a> are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure <a href="#org48a16fd">6</a>).
+The two sensors presented in Section <a href="#org3cd5225">1</a> are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure <a href="#orgffed4ec">6</a>).
 </p>
 
 
-<div id="org48a16fd" class="figure">
+<div id="orgffed4ec" class="figure">
 <p><img src="figs-paper/sensor_fusion_noise_arch.png" alt="sensor_fusion_noise_arch.png" />
 </p>
 <p><span class="figure-number">Figure 6: </span>Sensor Fusion Architecture</p>
@@ -423,11 +418,11 @@ The super sensor estimate \(\hat{x}\) is given by \eqref{eq:super_sensor_estimat
 </div>
 </div>
 
-<div id="outline-container-orge9e0bd4" class="outline-3">
-<h3 id="orge9e0bd4"><span class="section-number-3">2.2</span> Super Sensor Noise</h3>
+<div id="outline-container-org3a4f4b8" class="outline-3">
+<h3 id="org3a4f4b8"><span class="section-number-3">2.2</span> Super Sensor Noise</h3>
 <div class="outline-text-3" id="text-2-2">
 <p>
-<a id="orgff055a3"></a>
+<a id="org3e73362"></a>
 </p>
 
 <p>
@@ -458,15 +453,15 @@ And the Root Mean Square (RMS) value of the super sensor noise \(\sigma_n\) is g
 </div>
 </div>
 
-<div id="outline-container-orgefb4347" class="outline-3">
-<h3 id="orgefb4347"><span class="section-number-3">2.3</span> Super Sensor Dynamical Uncertainty</h3>
+<div id="outline-container-org3f0594e" class="outline-3">
+<h3 id="org3f0594e"><span class="section-number-3">2.3</span> Super Sensor Dynamical Uncertainty</h3>
 <div class="outline-text-3" id="text-2-3">
 <p>
-<a id="org859b213"></a>
+<a id="orgd73f82f"></a>
 </p>
 
 <p>
-If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure <a href="#org54538d7">7</a>), the super sensor dynamics is then equals to:
+If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure <a href="#org05e7617">7</a>), the super sensor dynamics is then equals to:
 </p>
 
 \begin{equation}
@@ -478,18 +473,18 @@ If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not
 \end{equation}
 
 
-<div id="org54538d7" class="figure">
+<div id="org05e7617" class="figure">
 <p><img src="figs-paper/sensor_model_uncertainty.png" alt="sensor_model_uncertainty.png" />
 </p>
 <p><span class="figure-number">Figure 7: </span>Sensor Model including Dynamical Uncertainty</p>
 </div>
 
 <p>
-The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure <a href="#orgb11ef23">8</a>.
+The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure <a href="#org1ff6ff8">8</a>.
 </p>
 
 
-<div id="orgb11ef23" class="figure">
+<div id="org1ff6ff8" class="figure">
 <p><img src="figs-paper/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
 </p>
 <p><span class="figure-number">Figure 8: </span>Super Sensor model uncertainty displayed in the complex plane</p>
@@ -498,11 +493,11 @@ The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at freque
 </div>
 </div>
 
-<div id="outline-container-org5896b60" class="outline-2">
-<h2 id="org5896b60"><span class="section-number-2">3</span> Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</h2>
+<div id="outline-container-org096192f" class="outline-2">
+<h2 id="org096192f"><span class="section-number-2">3</span> Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</h2>
 <div class="outline-text-2" id="text-3">
 <p>
-<a id="orgdd6b9ce"></a>
+<a id="orgbaf748a"></a>
 </p>
 <p>
 In this section, the complementary filters \(H_1(s)\) and \(H_2(s)\) are designed in order to minimize the RMS value of super sensor noise \(\sigma_n\).
@@ -520,23 +515,23 @@ The RMS value of the super sensor noise is (neglecting the model uncertainty):
 
 <p>
 The goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) (complementary property) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized (minimized RMS value of the super sensor noise).
-This is done using the \(\mathcal{H}_2\) synthesis in Section <a href="#org5bc9386">3.1</a>.
+This is done using the \(\mathcal{H}_2\) synthesis in Section <a href="#orgf1e20ae">3.1</a>.
 </p>
 </div>
 
-<div id="outline-container-orgf5a8a84" class="outline-3">
-<h3 id="orgf5a8a84"><span class="section-number-3">3.1</span> \(\mathcal{H}_2\) Synthesis</h3>
+<div id="outline-container-org9188eae" class="outline-3">
+<h3 id="org9188eae"><span class="section-number-3">3.1</span> \(\mathcal{H}_2\) Synthesis</h3>
 <div class="outline-text-3" id="text-3-1">
 <p>
-<a id="org5bc9386"></a>
+<a id="orgf1e20ae"></a>
 </p>
 
 <p>
-Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure <a href="#orgd218886">9</a> and described by Equation \eqref{eq:H2_generalized_plant}.
+Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure <a href="#org95769ad">9</a> and described by Equation \eqref{eq:H2_generalized_plant}.
 </p>
 
 
-<div id="orgd218886" class="figure">
+<div id="org95769ad" class="figure">
 <p><img src="figs-paper/h_two_optimal_fusion.png" alt="h_two_optimal_fusion.png" />
 </p>
 <p><span class="figure-number">Figure 9: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
@@ -592,10 +587,10 @@ Finally, \(H_1(s)\) is defined as follows
 </div>
 
 <p>
-The obtained complementary filters are shown in Figure <a href="#org8c7ba6b">10</a>.
+The obtained complementary filters are shown in Figure <a href="#org02514a9">10</a>.
 </p>
 
-<div id="org8c7ba6b" class="figure">
+<div id="org02514a9" class="figure">
 <p><img src="figs/htwo_comp_filters.png" alt="htwo_comp_filters.png" />
 </p>
 <p><span class="figure-number">Figure 10: </span>Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis</p>
@@ -603,11 +598,11 @@ The obtained complementary filters are shown in Figure <a href="#org8c7ba6b">10<
 </div>
 </div>
 
-<div id="outline-container-org72159df" class="outline-3">
-<h3 id="org72159df"><span class="section-number-3">3.2</span> Super Sensor Noise</h3>
+<div id="outline-container-org6d6d909" class="outline-3">
+<h3 id="org6d6d909"><span class="section-number-3">3.2</span> Super Sensor Noise</h3>
 <div class="outline-text-3" id="text-3-2">
 <p>
-<a id="orgc7cc0a8"></a>
+<a id="org497847a"></a>
 </p>
 
 <p>
@@ -622,13 +617,13 @@ PSD_H2 = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <span
 </div>
 
 <p>
-The obtained ASD are shown in Figure <a href="#orge481bcd">11</a>.
+The obtained ASD are shown in Figure <a href="#orgd3690df">11</a>.
 </p>
 
 <p>
-The RMS value of the individual sensors and of the super sensor are listed in Table <a href="#org3918d27">3</a>.
+The RMS value of the individual sensors and of the super sensor are listed in Table <a href="#org04c5321">3</a>.
 </p>
-<table id="org3918d27" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org04c5321" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 3:</span> RMS value of the individual sensor noise and of the super sensor using the \(\mathcal{H}_2\) Synthesis</caption>
 
 <colgroup>
@@ -661,7 +656,7 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
 </table>
 
 
-<div id="orge481bcd" class="figure">
+<div id="orgd3690df" class="figure">
 <p><img src="figs/psd_sensors_htwo_synthesis.png" alt="psd_sensors_htwo_synthesis.png" />
 </p>
 <p><span class="figure-number">Figure 11: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal</p>
@@ -670,19 +665,19 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
 <p>
 A time domain simulation is now performed.
 The measured velocity \(x\) is set to be a sweep sine with an amplitude of \(0.1\ [m/s]\).
-The velocity estimates from the two sensors and from the super sensors are shown in Figure <a href="#org9c2a7e4">12</a>.
-The resulting noises are displayed in Figure <a href="#org4cc42bf">13</a>.
+The velocity estimates from the two sensors and from the super sensors are shown in Figure <a href="#orgf112864">12</a>.
+The resulting noises are displayed in Figure <a href="#org1ab022b">13</a>.
 </p>
 
 
-<div id="org9c2a7e4" class="figure">
+<div id="orgf112864" class="figure">
 <p><img src="figs/super_sensor_time_domain_h2.png" alt="super_sensor_time_domain_h2.png" />
 </p>
 <p><span class="figure-number">Figure 12: </span>Noise of individual sensors and noise of the super sensor</p>
 </div>
 
 
-<div id="org4cc42bf" class="figure">
+<div id="org1ab022b" class="figure">
 <p><img src="figs/sensor_noise_H2_time_domain.png" alt="sensor_noise_H2_time_domain.png" />
 </p>
 <p><span class="figure-number">Figure 13: </span>Noise of the two sensors \(n_1, n_2\) and noise of the super sensor \(n\)</p>
@@ -690,15 +685,15 @@ The resulting noises are displayed in Figure <a href="#org4cc42bf">13</a>.
 </div>
 </div>
 
-<div id="outline-container-org8ffba19" class="outline-3">
-<h3 id="org8ffba19"><span class="section-number-3">3.3</span> Discrepancy between sensor dynamics and model</h3>
+<div id="outline-container-org850d99e" class="outline-3">
+<h3 id="org850d99e"><span class="section-number-3">3.3</span> Discrepancy between sensor dynamics and model</h3>
 <div class="outline-text-3" id="text-3-3">
 <p>
-If we consider sensor dynamical uncertainty as explained in Section <a href="#orgf04439d">1.2</a>, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
+If we consider sensor dynamical uncertainty as explained in Section <a href="#org3943238">1.2</a>, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
 </p>
 
 <p>
-The super sensor dynamical uncertainty is shown in Figure <a href="#org865879b">14</a>.
+The super sensor dynamical uncertainty is shown in Figure <a href="#org35f4dc2">14</a>.
 </p>
 
 <p>
@@ -706,7 +701,7 @@ It is shown that the phase uncertainty is not bounded between 100Hz and 200Hz.
 As a result the super sensor signal can not be used for feedback applications about 100Hz.
 </p>
 
-<div id="org865879b" class="figure">
+<div id="org35f4dc2" class="figure">
 <p><img src="figs/super_sensor_dynamical_uncertainty_H2.png" alt="super_sensor_dynamical_uncertainty_H2.png" />
 </p>
 <p><span class="figure-number">Figure 14: </span>Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis</p>
@@ -715,11 +710,11 @@ As a result the super sensor signal can not be used for feedback applications ab
 </div>
 </div>
 
-<div id="outline-container-org26ea7b1" class="outline-2">
-<h2 id="org26ea7b1"><span class="section-number-2">4</span> Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</h2>
+<div id="outline-container-org6d1270a" class="outline-2">
+<h2 id="org6d1270a"><span class="section-number-2">4</span> Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</h2>
 <div class="outline-text-2" id="text-4">
 <p>
-<a id="org5d93f37"></a>
+<a id="orgea66128"></a>
 </p>
 <p>
 We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
@@ -727,18 +722,18 @@ We initially considered perfectly known sensor dynamics so that it can be perfec
 
 <p>
 We now take into account the fact that the sensor dynamics is only partially known.
-To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure <a href="#org2b71ca6">15</a>.
+To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure <a href="#org3a26ad3">15</a>.
 </p>
 
 
-<div id="org2b71ca6" class="figure">
+<div id="org3a26ad3" class="figure">
 <p><img src="figs-paper/sensor_fusion_arch_uncertainty.png" alt="sensor_fusion_arch_uncertainty.png" />
 </p>
 <p><span class="figure-number">Figure 15: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
 </div>
 
 <p>
-As explained in Section <a href="#orgf04439d">1.2</a>, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
+As explained in Section <a href="#org3943238">1.2</a>, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
 </p>
 
 <p>
@@ -760,7 +755,7 @@ In order to specify a wanted upper bound on the dynamical uncertainty, a weight
 \end{align}
 
 <p>
-The choice of \(W_u\) is presented in Section <a href="#orgefed264">4.1</a>.
+The choice of \(W_u\) is presented in Section <a href="#org5b00e7a">4.1</a>.
 </p>
 
 
@@ -778,15 +773,15 @@ The objective is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s)
 </p>
 
 <p>
-This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org2c990ce">4.2</a>.
+This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org7bd80be">4.2</a>.
 </p>
 </div>
 
-<div id="outline-container-org8dfd9d2" class="outline-3">
-<h3 id="org8dfd9d2"><span class="section-number-3">4.1</span> Weighting Function used to bound the super sensor uncertainty</h3>
+<div id="outline-container-orgc5a0221" class="outline-3">
+<h3 id="orgc5a0221"><span class="section-number-3">4.1</span> Weighting Function used to bound the super sensor uncertainty</h3>
 <div class="outline-text-3" id="text-4-1">
 <p>
-<a id="orgefed264"></a>
+<a id="org5b00e7a"></a>
 </p>
 
 <p>
@@ -799,7 +794,7 @@ This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org
 \end{align}
 
 <p>
-The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure <a href="#orgdffee80">16</a>.
+The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure <a href="#orge13d408">16</a>.
 </p>
 
 <div class="org-src-container">
@@ -810,7 +805,7 @@ Wu = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
 </div>
 
 
-<div id="orgdffee80" class="figure">
+<div id="orge13d408" class="figure">
 <p><img src="figs/weight_uncertainty_bounds_Wu.png" alt="weight_uncertainty_bounds_Wu.png" />
 </p>
 <p><span class="figure-number">Figure 16: </span>Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)</p>
@@ -818,19 +813,19 @@ Wu = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
 </div>
 </div>
 
-<div id="outline-container-org7422ade" class="outline-3">
-<h3 id="org7422ade"><span class="section-number-3">4.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
+<div id="outline-container-org334f826" class="outline-3">
+<h3 id="org334f826"><span class="section-number-3">4.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
 <div class="outline-text-3" id="text-4-2">
 <p>
-<a id="org2c990ce"></a>
+<a id="org7bd80be"></a>
 </p>
 
 <p>
-The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure <a href="#org59777f5">17</a> and is described by Equation \eqref{eq:Hinf_generalized_plant}.
+The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure <a href="#org461ce6a">17</a> and is described by Equation \eqref{eq:Hinf_generalized_plant}.
 </p>
 
 
-<div id="org59777f5" class="figure">
+<div id="org461ce6a" class="figure">
 <p><img src="figs-paper/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
 </p>
 <p><span class="figure-number">Figure 17: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
@@ -866,7 +861,7 @@ And the \(\mathcal{H}_\infty\) synthesis is performed using the <code>hinfsyn</c
 </pre>
 </div>
 
-<pre class="example">
+<pre class="example" id="orge32bb33">
 Test bounds:  0.7071 &lt;=  gamma  &lt;=  1.291
 
   gamma        X&gt;=0        Y&gt;=0       rho(XY)&lt;1    p/f
@@ -897,11 +892,11 @@ The \(\mathcal{H}_\infty\) is successful as the \(\mathcal{H}_\infty\) norm of t
 </div>
 
 <p>
-The obtained complementary filters as well as the wanted upper bounds are shown in Figure <a href="#orga1806e3">18</a>.
+The obtained complementary filters as well as the wanted upper bounds are shown in Figure <a href="#org938a541">18</a>.
 </p>
 
 
-<div id="orga1806e3" class="figure">
+<div id="org938a541" class="figure">
 <p><img src="figs/hinf_comp_filters.png" alt="hinf_comp_filters.png" />
 </p>
 <p><span class="figure-number">Figure 18: </span>Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis</p>
@@ -909,11 +904,11 @@ The obtained complementary filters as well as the wanted upper bounds are shown
 </div>
 </div>
 
-<div id="outline-container-orga0267c6" class="outline-3">
-<h3 id="orga0267c6"><span class="section-number-3">4.3</span> Super sensor uncertainty</h3>
+<div id="outline-container-org211e432" class="outline-3">
+<h3 id="org211e432"><span class="section-number-3">4.3</span> Super sensor uncertainty</h3>
 <div class="outline-text-3" id="text-4-3">
 <p>
-The super sensor dynamical uncertainty is displayed in Figure <a href="#orge75f5ef">19</a>.
+The super sensor dynamical uncertainty is displayed in Figure <a href="#org52cdb78">19</a>.
 It is confirmed that the super sensor dynamical uncertainty is less than the maximum allowed uncertainty defined by the norm of \(W_u(s)\).
 </p>
 
@@ -922,7 +917,7 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
 </p>
 
 
-<div id="orge75f5ef" class="figure">
+<div id="org52cdb78" class="figure">
 <p><img src="figs/super_sensor_dynamical_uncertainty_Hinf.png" alt="super_sensor_dynamical_uncertainty_Hinf.png" />
 </p>
 <p><span class="figure-number">Figure 19: </span>Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis</p>
@@ -930,12 +925,12 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
 </div>
 </div>
 
-<div id="outline-container-org979fede" class="outline-3">
-<h3 id="org979fede"><span class="section-number-3">4.4</span> Super sensor noise</h3>
+<div id="outline-container-org7672e0e" class="outline-3">
+<h3 id="org7672e0e"><span class="section-number-3">4.4</span> Super sensor noise</h3>
 <div class="outline-text-3" id="text-4-4">
 <p>
 We now compute the obtain Power Spectral Density of the super sensor&rsquo;s noise.
-The Amplitude Spectral Densities are shown in Figure <a href="#org5aac73f">20</a>.
+The Amplitude Spectral Densities are shown in Figure <a href="#org06b08df">20</a>.
 </p>
 
 <div class="org-src-container">
@@ -947,18 +942,18 @@ PSD_Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <spa
 </div>
 
 <p>
-The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table <a href="#org3e08452">4</a>.
+The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table <a href="#org3cf92ce">4</a>.
 As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis is much noisier than the super sensor obtained from the \(\mathcal{H}_2\) synthesis.
 </p>
 
 
-<div id="org5aac73f" class="figure">
+<div id="org06b08df" class="figure">
 <p><img src="figs/psd_sensors_hinf_synthesis.png" alt="psd_sensors_hinf_synthesis.png" />
 </p>
-<p><span class="figure-number">Figure 20: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the</p>
+<p><span class="figure-number">Figure 20: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the \(\mathcal{H}_\infty\) synthesis</p>
 </div>
 
-<table id="org3e08452" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org3cf92ce" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 4:</span> Comparison of the obtained RMS noise of the super sensor</caption>
 
 <colgroup>
@@ -987,8 +982,8 @@ As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis
 </div>
 </div>
 
-<div id="outline-container-orgda33992" class="outline-3">
-<h3 id="orgda33992"><span class="section-number-3">4.5</span> Conclusion</h3>
+<div id="outline-container-orgdcb47cf" class="outline-3">
+<h3 id="orgdcb47cf"><span class="section-number-3">4.5</span> Conclusion</h3>
 <div class="outline-text-3" id="text-4-5">
 <p>
 Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values.
@@ -1001,22 +996,22 @@ However, the RMS of the super sensor noise is not optimized as it was the case w
 </div>
 </div>
 
-<div id="outline-container-org15afe90" class="outline-2">
-<h2 id="org15afe90"><span class="section-number-2">5</span> Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</h2>
+<div id="outline-container-org49a7cc2" class="outline-2">
+<h2 id="org49a7cc2"><span class="section-number-2">5</span> Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</h2>
 <div class="outline-text-2" id="text-5">
 <p>
-<a id="org9f98c16"></a>
+<a id="org2aeafb3"></a>
 </p>
 <p>
 The (optima) \(\mathcal{H}_2\) synthesis and the (robust) \(\mathcal{H}_\infty\) synthesis are now combined to form an Optimal and Robust synthesis of complementary filters for sensor fusion.
 </p>
 
 <p>
-The sensor fusion architecture is shown in Figure <a href="#org3cc874e">21</a> (\(\hat{G}_i\) are omitted for space reasons).
+The sensor fusion architecture is shown in Figure <a href="#orgd633d12">21</a> (\(\hat{G}_i\) are omitted for space reasons).
 </p>
 
 
-<div id="org3cc874e" class="figure">
+<div id="orgd633d12" class="figure">
 <p><img src="figs-paper/sensor_fusion_arch_full.png" alt="sensor_fusion_arch_full.png" />
 </p>
 <p><span class="figure-number">Figure 21: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
@@ -1031,18 +1026,18 @@ The goal is to design complementary filters such that:
 </ul>
 
 <p>
-To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section <a href="#orgbbc8594">5.1</a>.
+To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section <a href="#org669093b">5.1</a>.
 </p>
 </div>
-<div id="outline-container-org0f81a91" class="outline-3">
-<h3 id="org0f81a91"><span class="section-number-3">5.1</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</h3>
+<div id="outline-container-org7ed8ed7" class="outline-3">
+<h3 id="org7ed8ed7"><span class="section-number-3">5.1</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</h3>
 <div class="outline-text-3" id="text-5-1">
 <p>
-<a id="orgbbc8594"></a>
+<a id="org669093b"></a>
 </p>
 
 <p>
-The synthesis architecture that is used here is shown in Figure <a href="#orga971cdb">22</a>.
+The synthesis architecture that is used here is shown in Figure <a href="#orgc870005">22</a>.
 </p>
 
 <p>
@@ -1054,7 +1049,7 @@ The filter \(H_2(s)\) is synthesized such that it:
 </ul>
 
 
-<div id="orga971cdb" class="figure">
+<div id="orgc870005" class="figure">
 <p><img src="figs-paper/mixed_h2_hinf_synthesis.png" alt="mixed_h2_hinf_synthesis.png" />
 </p>
 <p><span class="figure-number">Figure 22: </span>Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
@@ -1098,11 +1093,11 @@ And the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed.
 </div>
 
 <p>
-The obtained complementary filters are shown in Figure <a href="#org30e8a3f">23</a>.
+The obtained complementary filters are shown in Figure <a href="#orgf37f308">23</a>.
 </p>
 
 
-<div id="org30e8a3f" class="figure">
+<div id="orgf37f308" class="figure">
 <p><img src="figs/htwo_hinf_comp_filters.png" alt="htwo_hinf_comp_filters.png" />
 </p>
 <p><span class="figure-number">Figure 23: </span>Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
@@ -1110,19 +1105,19 @@ The obtained complementary filters are shown in Figure <a href="#org30e8a3f">23<
 </div>
 </div>
 
-<div id="outline-container-org417aabd" class="outline-3">
-<h3 id="org417aabd"><span class="section-number-3">5.2</span> Obtained Super Sensor&rsquo;s noise</h3>
+<div id="outline-container-orgfb839cf" class="outline-3">
+<h3 id="orgfb839cf"><span class="section-number-3">5.2</span> Obtained Super Sensor&rsquo;s noise</h3>
 <div class="outline-text-3" id="text-5-2">
 <p>
-The Amplitude Spectral Density of the super sensor&rsquo;s noise is shown in Figure <a href="#orgfdfbf76">24</a>.
+The Amplitude Spectral Density of the super sensor&rsquo;s noise is shown in Figure <a href="#orgb3c92e4">24</a>.
 </p>
 
 <p>
-A time domain simulation is shown in Figure <a href="#orgdf5905f">25</a>.
+A time domain simulation is shown in Figure <a href="#org4a0db9b">25</a>.
 </p>
 
 <p>
-The RMS values of the super sensor noise for the presented three synthesis are listed in Table <a href="#org3b7df55">5</a>.
+The RMS values of the super sensor noise for the presented three synthesis are listed in Table <a href="#org3b10edf">5</a>.
 </p>
 
 <div class="org-src-container">
@@ -1134,20 +1129,20 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <s
 </div>
 
 
-<div id="orgfdfbf76" class="figure">
+<div id="orgb3c92e4" class="figure">
 <p><img src="figs/psd_sensors_htwo_hinf_synthesis.png" alt="psd_sensors_htwo_hinf_synthesis.png" />
 </p>
 <p><span class="figure-number">Figure 24: </span>Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
 </div>
 
 
-<div id="orgdf5905f" class="figure">
+<div id="org4a0db9b" class="figure">
 <p><img src="figs/super_sensor_time_domain_h2_hinf.png" alt="super_sensor_time_domain_h2_hinf.png" />
 </p>
 <p><span class="figure-number">Figure 25: </span>Noise of individual sensors and noise of the super sensor</p>
 </div>
 
-<table id="org3b7df55" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org3b10edf" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 5:</span> Comparison of the obtained RMS noise of the super sensor</caption>
 
 <colgroup>
@@ -1181,15 +1176,15 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <s
 </div>
 </div>
 
-<div id="outline-container-org2dce888" class="outline-3">
-<h3 id="org2dce888"><span class="section-number-3">5.3</span> Obtained Super Sensor&rsquo;s Uncertainty</h3>
+<div id="outline-container-orgb8596ac" class="outline-3">
+<h3 id="orgb8596ac"><span class="section-number-3">5.3</span> Obtained Super Sensor&rsquo;s Uncertainty</h3>
 <div class="outline-text-3" id="text-5-3">
 <p>
-The uncertainty on the super sensor&rsquo;s dynamics is shown in Figure <a href="#orgb2d28c5">26</a>.
+The uncertainty on the super sensor&rsquo;s dynamics is shown in Figure <a href="#org11fe014">26</a>.
 </p>
 
 
-<div id="orgb2d28c5" class="figure">
+<div id="org11fe014" class="figure">
 <p><img src="figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.png" alt="super_sensor_dynamical_uncertainty_Htwo_Hinf.png" />
 </p>
 <p><span class="figure-number">Figure 26: </span>Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
@@ -1197,8 +1192,8 @@ The uncertainty on the super sensor&rsquo;s dynamics is shown in Figure <a href=
 </div>
 </div>
 
-<div id="outline-container-org47da78c" class="outline-3">
-<h3 id="org47da78c"><span class="section-number-3">5.4</span> Conclusion</h3>
+<div id="outline-container-orgfcfa8a2" class="outline-3">
+<h3 id="orgfcfa8a2"><span class="section-number-3">5.4</span> Conclusion</h3>
 <div class="outline-text-3" id="text-5-4">
 <p>
 The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary filters allows to:
@@ -1211,18 +1206,18 @@ The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary fi
 </div>
 </div>
 
-<div id="outline-container-org0afe5ef" class="outline-2">
-<h2 id="org0afe5ef"><span class="section-number-2">6</span> Matlab Functions</h2>
+<div id="outline-container-orgf267700" class="outline-2">
+<h2 id="orgf267700"><span class="section-number-2">6</span> Matlab Functions</h2>
 <div class="outline-text-2" id="text-6">
 <p>
-<a id="orgf41dc8d"></a>
+<a id="org50a28ec"></a>
 </p>
 </div>
-<div id="outline-container-orge81e522" class="outline-3">
-<h3 id="orge81e522"><span class="section-number-3">6.1</span> <code>createWeight</code></h3>
+<div id="outline-container-org005db82" class="outline-3">
+<h3 id="org005db82"><span class="section-number-3">6.1</span> <code>createWeight</code></h3>
 <div class="outline-text-3" id="text-6-1">
 <p>
-<a id="org89bad6d"></a>
+<a id="orgb602691"></a>
 </p>
 
 <p>
@@ -1274,11 +1269,11 @@ This Matlab function is accessible <a href="src/createWeight.m">here</a>.
 </div>
 </div>
 
-<div id="outline-container-org37ec2b4" class="outline-3">
-<h3 id="org37ec2b4"><span class="section-number-3">6.2</span> <code>plotMagUncertainty</code></h3>
+<div id="outline-container-org816b32a" class="outline-3">
+<h3 id="org816b32a"><span class="section-number-3">6.2</span> <code>plotMagUncertainty</code></h3>
 <div class="outline-text-3" id="text-6-2">
 <p>
-<a id="org8739875"></a>
+<a id="org2963b0d"></a>
 </p>
 
 <p>
@@ -1329,11 +1324,11 @@ p.FaceAlpha = args.opacity;
 </div>
 </div>
 
-<div id="outline-container-org9f73572" class="outline-3">
-<h3 id="org9f73572"><span class="section-number-3">6.3</span> <code>plotPhaseUncertainty</code></h3>
+<div id="outline-container-orgddb3773" class="outline-3">
+<h3 id="orgddb3773"><span class="section-number-3">6.3</span> <code>plotPhaseUncertainty</code></h3>
 <div class="outline-text-3" id="text-6-3">
 <p>
-<a id="org49e21eb"></a>
+<a id="org8d340da"></a>
 </p>
 
 <p>
@@ -1397,7 +1392,7 @@ p.FaceAlpha = args.opacity;
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Thomas Dehaeze</p>
-<p class="date">Created: 2020-10-05 lun. 15:08</p>
+<p class="date">Created: 2020-11-12 jeu. 10:42</p>
 </div>
 </body>
 </html>

paper/config.tex

@@ -1,28 +0,0 @@
-% H Infini
-\newcommand{\hinf}{\mathcal{H}_\infty}
-
-% H 2
-\newcommand{\htwo}{\mathcal{H}_2}
-
-% Omega
-\newcommand{\w}{\omega}
-
-% H-Infinity Norm
-\newcommand{\hnorm}[1]{\left\|#1\right\|_{\infty}}
-
-% H-2 Norm
-\newcommand{\normtwo}[1]{\left\|#1\right\|_{2}}
-
-% Norm
-\newcommand{\norm}[1]{\left\|#1\right\|}
-
-% Absolute value
-\newcommand{\abs}[1]{\left\lvert #1 \right\lvert}
-
-% Minimum Subscript
-\newcommand{\smin}{_{\text{min}}}
-
-% Maximum Subscript
-\newcommand{\smax}{_{\text{max}}}
-
-\newcommand*\colvec[1]{\begin{bmatrix}#1\end{bmatrix}}