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>
+<title>Robust and Optimal Sensor Fusion using Complementary Filters</title>
-<meta name="generator" content="Org mode" />
 <meta name="author" content="Thomas Dehaeze" />
-<link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
+<meta name="generator" content="Org Mode" />
-<link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
+<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
-<script src="js/jquery.min.js"></script>
+<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
-<script src="js/bootstrap.min.js"></script>
+<style> #content {margin: auto;} </style>
-<script src="js/jquery.stickytableheaders.min.js"></script>
-<script src="js/readtheorg.js"></script>
 </head>
 <body>
 <div id="org-div-home-and-up">
  <a accesskey="h" href="../index.html"> UP </a>
  |
  <a accesskey="H" href="../index.html"> HOME </a>
-</div><div id="content">
+</div><div id="content" class="content">
-<h1 class="title">Robust and Optimal Sensor Fusion
+<h1 class="title">Robust and Optimal Sensor Fusion using Complementary Filters
 <br />
 <span class="subtitle">Dehaeze Thomas and Collette Christophe</span>
 </h1>
-
+<blockquote>
-<div id="outline-container-org0ec4732" class="outline-2">
-<h2 id="org0ec4732">Paper</h2>
-<div class="outline-text-2" id="text-org0ec4732">
 <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">
+<div id="outline-container-orgd50e329" class="outline-2">
-<h2 id="orge418504">Matlab Scripts</h2>
+<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-orge418504">
+<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">
+<div id="outline-container-org3b999bd" class="outline-2">
-<h2 id="org99adac0">Tikz Figures</h2>
+<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-org99adac0">
+<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:\author{Thomas~Dehaeze, Mohit~Verma, and~Christophe~Collette@@
-#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\@@
+#+AUTHOR: @@latex:\thanks{The authors are with the Precision Mechatronics Laboratory at the University of Liege, Belgium.}@@
-#+AUTHOR: @@latex:Grenoble, France\\@@
+#+AUTHOR: @@latex:\thanks{Corresponding author: thomas.dehaeze@esrf.fr}@@
-#+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:}@@
 
 #+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
+(add-to-list 'org-latex-classes
-               '("IEEEtran"
+             '("IEEEtran"
-                 "\\documentclass{IEEEtran}"
+               "\\documentclass{IEEEtran}"
-                 ("\\section{%s}" . "\\section*{%s}")
+               ("\\section{%s}" . "\\section*{%s}")
-                 ("\\subsection{%s}" . "\\subsection*{%s}")
+               ("\\subsection{%s}" . "\\subsection*{%s}")
-                 ("\\subsubsection{%s}" . "\\subsubsection*{%s}")
+               ("\\subsubsection{%s}" . "\\subsubsection*{%s}")
-                 ("\\paragraph{%s}" . "\\paragraph*{%s}")
+               ("\\paragraph{%s}" . "\\paragraph*{%s}")
-                 ("\\subparagraph{%s}" . "\\subparagraph*{%s}"))
+               ("\\subparagraph{%s}" . "\\subparagraph*{%s}"))
-               )
+             )
 
-  (defun delete-org-comments (backend)
+;; Remove automatic org headings
-    (loop for comment in (reverse (org-element-map (org-element-parse-buffer)
+(defun my-latex-filter-removeOrgAutoLabels (text backend info)
-                                      'comment 'identity))
+  "Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings."
-          do
+  (when (org-export-derived-backend-p backend 'latex)
-          (setf (buffer-substring (org-element-property :begin comment)
+    (replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text)))
-                                  (org-element-property :end comment))
-                "")))
 
-  ;; add to export hook
+(add-to-list 'org-export-filter-headline-functions
-  (add-hook 'org-export-before-processing-hook 'delete-org-comments)
+             'my-latex-filter-removeOrgAutoLabels)
 
-  ;; Remove hypersetup
+;; Automatic delete org org-comments
-  (setq org-latex-with-hyperref nil)
+(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
@@ -96,7 +92,6 @@
 <<sec:optimal_fusion>>
 
 ** Sensor Model
-
 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 $G_i(s)$.
 
@@ -131,7 +126,6 @@ In order to obtain an estimate $\hat{x}_i$ of $x$, a model $\hat{G}_i$ of the (t
 [[file:figs/sensor_model_noise.pdf]]
 
 ** Sensor Fusion Architecture
-
 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.
@@ -195,8 +189,8 @@ This can be cast into an $\mathcal{H}_2$ synthesis problem by considering the fo
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
 \end{pmatrix} = \underbrace{\begin{bmatrix}
-  N_1 & N_1 \\
+  N_1 & -N_1 \\
-  0   & N_2 \\
+  0   &  N_2 \\
   1   &  0
 \end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
   w \\ u
@@ -223,8 +217,46 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
 
 ** Example
 
+#+name: fig:sensors_nominal_dynamics
+#+caption: Sensor nominal dynamics from the velocity of the object to the output voltage
+#+attr_latex: :scale 1
+[[file:figs/sensors_nominal_dynamics.pdf]]
+
+#+name: fig:sensors_noise
+#+caption: Amplitude spectral density of the sensors $\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|$
+#+attr_latex: :scale 1
+[[file:figs/sensors_noise.pdf]]
+
+
+#+name: fig:htwo_comp_filters
+#+caption:  Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/htwo_comp_filters.pdf]]
+
+
+#+name: fig:psd_sensors_htwo_synthesis
+#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
+#+attr_latex: :scale 1
+[[file:figs/psd_sensors_htwo_synthesis.pdf]]
+
+
+#+name: fig:super_sensor_time_domain_h2
+#+caption: Noise of individual sensors and noise of the super sensor
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_time_domain_h2.pdf]]
+
 ** Robustness Problem
 
+#+name: fig:sensors_nominal_dynamics_and_uncertainty
+#+caption: Nominal Sensor Dynamics $\hat{G}_i$ (solid lines) as well as the spread of the dynamical uncertainty (background color)
+#+attr_latex: :scale 1
+[[file:figs/sensors_nominal_dynamics_and_uncertainty.pdf]]
+
+#+name: fig:super_sensor_dynamical_uncertainty_H2
+#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_dynamical_uncertainty_H2.pdf]]
+
 * Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
 <<sec:robust_fusion>>
 
@@ -270,7 +302,6 @@ As $H_1$ and $H_2$ are complementary filters, we finally have:
 [[file:figs/sensor_fusion_arch_uncertainty.pdf]]
 
 ** Super Sensor Dynamical Uncertainty
-
 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.
 
 
@@ -306,9 +337,9 @@ This problem can thus be dealt with an $\mathcal{H}_\infty$ synthesis problem by
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
 \end{pmatrix} = \underbrace{\begin{bmatrix}
-  W_u W_1 & W_u W_1 \\
+  W_u W_1 & -W_u W_1 \\
-  0       & W_u W_2 \\
+  0       &  W_u W_2 \\
-  1       & 0
+  1       &  0
 \end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
   w \\ u
 \end{pmatrix}
@@ -332,6 +363,33 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
 
 ** Example
 
+#+name: fig:sensors_uncertainty_weights
+#+caption: Magnitude of the multiplicative uncertainty weights $|W_i(j\omega)|$
+#+attr_latex: :scale 1
+[[file:figs/sensors_uncertainty_weights.pdf]]
+
+
+#+name: fig:weight_uncertainty_bounds_Wu
+#+caption: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)
+#+attr_latex: :scale 1
+[[file:figs/weight_uncertainty_bounds_Wu.pdf]]
+
+#+name: fig:hinf_comp_filters
+#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/hinf_comp_filters.pdf]]
+
+#+name: fig:super_sensor_dynamical_uncertainty_Hinf
+#+caption: Super sensor dynamical uncertainty (solid curve) when using the $\mathcal{H}_\infty$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_dynamical_uncertainty_Hinf.pdf]]
+
+#+name: fig:psd_sensors_hinf_synthesis
+#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
+#+attr_latex: :scale 1
+[[file:figs/psd_sensors_hinf_synthesis.pdf]]
+
+
 * Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
 <<sec:optimal_robust_fusion>>
 
@@ -406,12 +464,32 @@ 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]]
 
 ** Example
 
+#+name: fig:htwo_hinf_comp_filters
+#+caption: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
+#+attr_latex: :scale 1
+[[file:figs/htwo_hinf_comp_filters.pdf]]
+
+#+name: fig:psd_sensors_htwo_hinf_synthesis
+#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
+#+attr_latex: :scale 1
+[[file:figs/psd_sensors_htwo_hinf_synthesis.pdf]]
+
+#+name: fig:super_sensor_time_domain_h2_hinf
+#+caption: Noise of individual sensors and noise of the super sensor
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_time_domain_h2_hinf.pdf]]
+
+#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf
+#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf]]
+
 * Experimental Validation
 <<sec:experimental_validation>>
 
@@ -429,4 +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 2020-09-23 mer. 14:15
+% 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 }}
+\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{2020-09-23}
+\date{2021-09-10}
-\title{Optimal and Robust Sensor Fusion}
+\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:org88afd51}
 \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}
@@ -61,12 +56,9 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
 \end{itemize}
 
 \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
-\label{sec:org5853545}
 \label{sec:optimal_fusion}
 
 \subsection{Sensor Model}
-\label{sec:org565ea86}
-
 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 \(G_i(s)\).
 
@@ -97,20 +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:org1ae73e8}
-
 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}
 
@@ -140,7 +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:orgb2e8dd6}
 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
@@ -154,7 +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:orga4cf5f1}
 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 \(n\):
@@ -170,8 +158,8 @@ This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
 \end{pmatrix} = \underbrace{\begin{bmatrix}
-  N_1 & N_1 \\
+  N_1 & -N_1 \\
-  0   & N_2 \\
+  0   &  N_2 \\
   1   &  0
 \end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
   w \\ u
@@ -193,22 +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:org74634c9}
+
+\begin{figure}[htbp]
+\centering
+\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,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,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,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,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:org5fda5c1}
+
+\begin{figure}[htbp]
+\centering
+\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,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:orgc88050f}
 \label{sec:robust_fusion}
 
 \subsection{Representation of Sensor Dynamical Uncertainty}
-\label{sec:orgb09aa5a}
 
 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.
@@ -223,12 +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:org1d92a74}
 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:
@@ -248,13 +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:org81db1d8}
-
 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}.
 
 
@@ -262,14 +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:org0e2a7a8}
 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.
@@ -279,7 +304,7 @@ To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such t
   \left| W_1(j\omega)H_1(j\omega) \right| + \left| W_2(j\omega)H_2(j\omega) \right| < \frac{1}{\left| W_u(j\omega) \right|}, \quad \forall \omega
 \end{equation}
 
-This is actually almost equivalent (to within a factor \(\sqrt{2}\)) equivalent as to have:
+This is actually almost equivalent as to have (within a factor \(\sqrt{2}\)):
 \begin{equation}
   \left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
 \end{equation}
@@ -289,9 +314,9 @@ This problem can thus be dealt with an \(\mathcal{H}_\infty\) synthesis problem
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
 \end{pmatrix} = \underbrace{\begin{bmatrix}
-  W_u W_1 & W_u W_1 \\
+  W_u W_1 & -W_u W_1 \\
-  0       & W_u W_2 \\
+  0       &  W_u W_2 \\
-  1       & 0
+  1       &  0
 \end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
   w \\ u
 \end{pmatrix}
@@ -310,71 +335,169 @@ 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:org0122000}
+
+\begin{figure}[htbp]
+\centering
+\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,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,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,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,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:orgdf5a196}
 \label{sec:optimal_robust_fusion}
 
-\subsection{Sensor Fusion Architecture}
+\subsection{Sensor with noise and model uncertainty}
-\label{sec:orge16b510}
+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 \(N_i(s)\).
+The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
+\begin{equation}
+  v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
+\end{equation}
+
+Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate \(\hat{x}_i\) of \(x\):
+\begin{equation}
+  \hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
+\end{equation}
 
 \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}
+
+For reason of space, the blocks \(\hat{G}_i\) and \(\hat{G}_i^{-1}\) are omitted.
+
+\begin{equation}
+\begin{aligned}
+  \hat{x} = &\Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x \\
+            &+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
+\end{aligned}
+\end{equation}
+
+\begin{equation}
+\begin{aligned}
+  \hat{x} = &\Big( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \Big) x \\
+            &+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
+\end{aligned}
+\end{equation}
+
+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{Synthesis Objective}
-\label{sec:orgb4b43b3}
-
 \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:orgb9b52ad}
+
+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.
+
+To specify how small we want the super sensor dynamic spread, we use a weighting filter \(W_u(s)\) as was done in Section \ref{sec:robust_fusion}.
+
+
+This synthesis problem can be solved using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis on the following generalized plant:
+\begin{equation}
+\begin{pmatrix}
+  z_{\infty, 1} \\ z_{\infty, 2} \\ z_{2, 1} \\ z_{2, 2} \\ v
+\end{pmatrix} = \underbrace{\begin{bmatrix}
+  W_u W_1 & W_u W_1 \\
+  0       & W_u W_2 \\
+  N_1     & N_1 \\
+  0       & N_2 \\
+  1       & 0
+\end{bmatrix}}_{P_{\mathcal{H}_2/\mathcal{H}_\infty}} \begin{pmatrix}
+  w \\ u
+\end{pmatrix}
+\end{equation}
+
+The synthesis objective is to:
+\begin{itemize}
+\item Keep the \(\mathcal{H}_\infty\) norm from \(w\) to \((z_{\infty,1}, z_{\infty,2})\) below \(1\)
+\item Minimize the \(\mathcal{H}_2\) norm from \(w\) to \((z_{2,1}, z_{2,2})\)
+\end{itemize}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/mixed_h2_hinf_synthesis.pdf}
+\includegraphics[scale=1,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}
+\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:orgc881f20}
+
+\begin{figure}[htbp]
+\centering
+\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,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,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,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:org05b79a0}
 \label{sec:experimental_validation}
 
 \subsection{Experimental Setup}
-\label{sec:orgc3daf35}
 
 \subsection{Sensor Noise and Dynamical Uncertainty}
-\label{sec:org26fedf6}
 
 \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:org72f2969}
 
 \subsection{Super Sensor Noise and Dynamical Uncertainty}
-\label{sec:orgf66f78b}
 
 \section{Conclusion}
-\label{sec:orge0f0a43}
 \label{sec:conclusion}
 
 \section{Acknowledgment}
-\label{sec:orgb16559e}
 
+\bibliographystyle{IEEEtran}
 \bibliography{ref}
 \end{document}

js/jquery.stickytableheaders.min.js

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js/readtheorg.js

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-$(function() {
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-    $('.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() {
-      $("[data-toggle='rst-versions']").toggleClass("shift-up");
-    });
-    // Make tables responsive
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-});
-
-$( 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>');
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-
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-window.SphinxRtdTheme = (function (jquery) {
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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_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>
 
 #+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
 
@@ -23,6 +18,10 @@
 #+LATEX_CLASS: cleanreport
 #+LATEX_CLASS_OPTIONS: [tocnp, minted, secbreak]
 
+#+LATEX_HEADER_EXTRA: \usepackage[cache=false]{minted}
+#+LATEX_HEADER_EXTRA: \usemintedstyle{autumn}
+#+LATEX_HEADER_EXTRA: \setminted[matlab]{linenos=true, breaklines=true, tabsize=4, fontsize=\scriptsize, autogobble=true}
+
 #+LATEX_HEADER: \newcommand{\authorFirstName}{Thomas}
 #+LATEX_HEADER: \newcommand{\authorLastName}{Dehaeze}
 #+LATEX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
@@ -71,13 +70,16 @@ In this example, the measured quantity $x$ is the velocity of an object.
 #+caption: Description of signals in Figure [[fig:sensor_model_noise_uncertainty]]
 #+attr_latex: :environment tabular :align clc
 #+attr_latex: :center t :booktabs t :float t
-| *Notation*    | *Meaning*                       | *Unit*  |
+| *Notation*       | *Meaning*                         | *Unit*                    |
-|---------------+---------------------------------+---------|
+|------------------+-----------------------------------+---------------------------|
-| $x$           | Physical measured quantity      | $[m/s]$ |
+| $x$              | Physical measured quantity        | $[m/s]$                   |
-| $\tilde{n}_i$ | White noise with unitary PSD    |         |
+| $\tilde{n}_i$    | White noise with unitary PSD      |                           |
-| $n_i$         | Shaped noise                    | $[m/s]$ |
+| $n_i$            | Shaped noise                      | $[m/s]$                   |
-| $v_i$         | Sensor output measurement       | $[V]$   |
+| $v_i$            | Sensor output measurement         | $[V]$                     |
-| $\hat{x}_i$   | Estimate of $x$ from the sensor | $[m/s]$ |
+| $\hat{x}_i$      | Estimate of $x$ from the sensor   | $[m/s]$                   |
+| $\Phi_n(\omega)$ | Power Spectral Density of $n$     | $[\frac{(m/s)^2}{Hz}]$    |
+| $\phi_n(\omega)$ | Amplitude Spectral Density of $n$ | $[\frac{m/s}{\sqrt{Hz}}]$ |
+| $\sigma_n$       | Root Mean Square Value of $n$     | $[m/s\ rms]$              |
 
 #+name: tab:sensor_dynamical_blocks
 #+caption: Description of Systems in Figure [[fig:sensor_model_noise_uncertainty]]
@@ -93,7 +95,7 @@ In this example, the measured quantity $x$ is the velocity of an object.
 #+name: fig:sensor_model_noise_uncertainty
 #+caption: Sensor Model
 #+RESULTS:
-[[file:figs-tikz/sensor_model_noise_uncertainty.png]]
+[[file:figs-paper/sensor_model_noise_uncertainty.png]]
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -138,18 +140,20 @@ 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', '$G_2(j\omega)$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Magnitude $[\frac{V}{m/s}]$'); set(gca, 'XTickLabel',[]);
+  ylabel('Magnitude $\left[\frac{V}{m/s}\right]$'); set(gca, 'XTickLabel',[]);
-  legend('location', 'northeast');
+  legend('location', 'northeast', 'FontSize', 8);
   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'))), '-');
@@ -158,12 +162,13 @@ 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
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/sensors_nominal_dynamics.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/sensors_nominal_dynamics.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:sensors_nominal_dynamics
@@ -183,7 +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);
@@ -201,11 +206,11 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
   xlabel('Frequency [Hz]'); ylabel('Magnitude');
   ylim([0, 5]);
   xlim([freqs(1), freqs(end)]);
-  legend('location', 'northwest');
+  legend('location', 'northwest', 'FontSize', 8);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/sensors_uncertainty_weights.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/sensors_uncertainty_weights.pdf', 'width', 'half', 'height', 'short');
 #+end_src
 
 #+name: fig:sensors_uncertainty_weights
@@ -215,8 +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', '$G_2$');
@@ -228,11 +235,12 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
   set(gca, 'XTickLabel',[]);
   ylabel('Magnitude $[\frac{V}{m/s}]$');
   ylim([1e-2, 2e3]);
-  legend('location', 'northeast');
+  legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2);
   hold off;
+  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);
@@ -245,12 +253,13 @@ 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
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/sensors_nominal_dynamics_and_uncertainty.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/sensors_nominal_dynamics_and_uncertainty.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:sensors_nominal_dynamics_and_uncertainty
@@ -285,14 +294,14 @@ The weights $N_1$ and $N_2$ representing the amplitude spectral density of the s
   plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1(j\omega)|$');
   plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2(j\omega)|$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Amplitude Spectral Density $\left[ \frac{m/s}{\sqrt{Hz}} \right]$');
+  xlabel('Frequency [Hz]'); ylabel('ASD $\left[ \frac{m/s}{\sqrt{Hz}} \right]$');
   hold off;
   xlim([freqs(1), freqs(end)]);
-  legend('location', 'northeast');
+  legend('location', 'northeast', 'FontSize', 8);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/sensors_noise.pdf', 'width', 'normal', 'height', 'normal');
+  exportFig('figs/sensors_noise.pdf', 'width', 'half', 'height', 'short');
 #+end_src
 
 #+name: fig:sensors_noise
@@ -317,7 +326,7 @@ The two sensors presented in Section [[sec:sensor_description]] are now merged t
 
 #+name: fig:sensor_fusion_noise_arch
 #+caption: Sensor Fusion Architecture
-[[file:figs-tikz/sensor_fusion_noise_arch.png]]
+[[file:figs-paper/sensor_fusion_noise_arch.png]]
 
 The complementary property of $H_1(s)$ and $H_2(s)$ means that the sum of their transfer function is equal to $1$ eqref:eq:complementary_property.
 
@@ -369,13 +378,13 @@ If we consider some dynamical uncertainty (the true system dynamics $G_i$ not be
 
 #+name: fig:sensor_model_uncertainty
 #+caption: Sensor Model including Dynamical Uncertainty
-[[file:figs-tikz/sensor_model_uncertainty.png]]
+[[file:figs-paper/sensor_model_uncertainty.png]]
 
 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 [[fig:uncertainty_set_super_sensor]].
 
 #+name: fig:uncertainty_set_super_sensor
 #+caption: Super Sensor model uncertainty displayed in the complex plane
-[[file:figs-tikz/uncertainty_set_super_sensor.png]]
+[[file:figs-paper/uncertainty_set_super_sensor.png]]
 
 * Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
 :PROPERTIES:
@@ -387,10 +396,6 @@ The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency
 ** Introduction                                                      :ignore:
 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 $\sigma_n$.
 
-#+name: fig:sensor_fusion_noise_arch
-#+caption: Optimal Sensor Fusion Architecture
-[[file:figs-tikz/sensor_fusion_noise_arch.png]]
-
 The RMS value of the super sensor noise is (neglecting the model uncertainty):
 \begin{equation}
 \begin{aligned}
@@ -423,7 +428,7 @@ Consider the generalized plant $P_{\mathcal{H}_2}$ shown in Figure [[fig:h_two_o
 
 #+name: fig:h_two_optimal_fusion
 #+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
-[[file:figs-tikz/h_two_optimal_fusion.png]]
+[[file:figs-paper/h_two_optimal_fusion.png]]
 
 \begin{equation} \label{eq:H2_generalized_plant}
 \begin{pmatrix}
@@ -469,9 +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', '$H_2$');
@@ -479,10 +485,10 @@ The obtained complementary filters are shown in Figure [[fig:htwo_comp_filters]]
   set(gca, 'XTickLabel',[]);
   ylabel('Magnitude');
   hold off;
-  legend('location', 'northeast');
+  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'))));
@@ -496,7 +502,7 @@ The obtained complementary filters are shown in Figure [[fig:htwo_comp_filters]]
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/htwo_comp_filters.pdf', 'width', 'full', 'height', 'tall');
+  exportFig('figs/htwo_comp_filters.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:htwo_comp_filters
@@ -507,7 +513,7 @@ The obtained complementary filters are shown in Figure [[fig:htwo_comp_filters]]
 ** Super Sensor Noise
 <<sec:H2_super_sensor_noise>>
 
-The Power Spectral Density of the individual sensors' noise $\Phi_{n_1}, \Phi_{n_2}$ and of the super sensor noise $\Phi_{n_{\mathcal{H}_2}}$ are computed below and shown in Figure [[fig:psd_sensors_htwo_synthesis]].
+The Power Spectral Density of the individual sensors' noise $\Phi_{n_1}, \Phi_{n_2}$ and of the super sensor noise $\Phi_{n_{\mathcal{H}_2}}$ are computed below.
 #+begin_src matlab
   PSD_S1 = abs(squeeze(freqresp(N1,    freqs, 'Hz'))).^2;
   PSD_S2 = abs(squeeze(freqresp(N2,    freqs, 'Hz'))).^2;
@@ -515,6 +521,8 @@ The Power Spectral Density of the individual sensors' noise $\Phi_{n_1}, \Phi_{n
            abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
 #+end_src
 
+The obtained ASD are shown in Figure [[fig:psd_sensors_htwo_synthesis]].
+
 The RMS value of the individual sensors and of the super sensor are listed in Table [[tab:rms_noise_H2]].
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
   data2orgtable([sqrt(trapz(freqs, PSD_S1)); sqrt(trapz(freqs, PSD_S2)); sqrt(trapz(freqs, PSD_H2))], ...
@@ -536,18 +544,18 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, PSD_S1, '-',  'DisplayName', '$\Phi_{n_1}$');
+  plot(freqs, sqrt(PSD_S1), '-',  'DisplayName', '$\phi_{n_1}$');
-  plot(freqs, PSD_S2, '-',  'DisplayName', '$\Phi_{n_2}$');
+  plot(freqs, sqrt(PSD_S2), '-',  'DisplayName', '$\phi_{n_2}$');
-  plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$');
+  plot(freqs, sqrt(PSD_H2), 'k-', 'DisplayName', '$\phi_{n_{\mathcal{H}_2}}$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density $\left[ \frac{(m/s)^2}{Hz} \right]$');
+  xlabel('Frequency [Hz]'); ylabel('ASD $\left[ \frac{m/s}{\sqrt{Hz}} \right]$');
   hold off;
   xlim([freqs(1), freqs(end)]);
-  legend('location', 'northeast');
+  legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/psd_sensors_htwo_synthesis.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/psd_sensors_htwo_synthesis.pdf', 'width', 'half', 'height', 'short');
 #+end_src
 
 #+name: fig:psd_sensors_htwo_synthesis
@@ -585,12 +593,12 @@ The resulting noises are displayed in Figure [[fig:sensor_noise_H2_time_domain]]
   plot(t, v, 'k--', 'DisplayName', '$x$');
   hold off;
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
-  legend('location', 'southwest');
+  legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
   ylim([-0.3, 0.3]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/super_sensor_time_domain_h2.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/super_sensor_time_domain_h2.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
 #+name: fig:super_sensor_time_domain_h2
@@ -609,12 +617,12 @@ The resulting noises are displayed in Figure [[fig:sensor_noise_H2_time_domain]]
   plot(t, (lsim(H1, n1, t)+lsim(H2, n2, t)), '-', 'DisplayName', '$n$');
   hold off;
   xlabel('Time [s]'); ylabel('Sensor Noise [m/s]');
-  legend();
+  legend('FontSize', 8);
   ylim([-0.2, 0.2]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/sensor_noise_H2_time_domain.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/sensor_noise_H2_time_domain.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
 #+name: fig:sensor_noise_H2_time_domain
@@ -634,8 +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', '$1 + W_2 \Delta_2$');
@@ -647,11 +657,11 @@ As a result the super sensor signal can not be used for feedback applications ab
   set(gca, 'XTickLabel',[]);
   ylabel('Magnitude');
   ylim([1e-2, 1e1]);
-  legend('location', 'southeast');
+  legend('location', 'southeast', 'FontSize', 8);
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -662,12 +672,13 @@ 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
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/super_sensor_dynamical_uncertainty_H2.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/super_sensor_dynamical_uncertainty_H2.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:super_sensor_dynamical_uncertainty_H2
@@ -690,7 +701,7 @@ To do so, we model the uncertainty that we have on the sensor dynamics by multip
 
 #+name: fig:sensor_fusion_arch_uncertainty
 #+caption: Sensor fusion architecture with sensor dynamics uncertainty
-[[file:figs-tikz/sensor_fusion_arch_uncertainty.png]]
+[[file:figs-paper/sensor_fusion_arch_uncertainty.png]]
 
 As explained in Section [[sec:sensor_uncertainty]], 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.
 
@@ -764,8 +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', '$1 + W_2 \Delta_2$');
@@ -777,11 +790,11 @@ The uncertainty bounds of the two individual sensor as well as the wanted maximu
   set(gca, 'XTickLabel',[]);
   ylabel('Magnitude');
   ylim([1e-2, 1e1]);
-  legend('location', 'southeast');
+  legend('location', 'southeast', 'FontSize', 8);
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -797,7 +810,7 @@ The uncertainty bounds of the two individual sensor as well as the wanted maximu
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/weight_uncertainty_bounds_Wu.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/weight_uncertainty_bounds_Wu.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:weight_uncertainty_bounds_Wu
@@ -812,7 +825,7 @@ The generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$
 
 #+name: fig:h_infinity_robust_fusion
 #+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
-[[file:figs-tikz/h_infinity_robust_fusion.png]]
+[[file:figs-paper/h_infinity_robust_fusion.png]]
 
 \begin{equation} \label{eq:Hinf_generalized_plant}
 \begin{pmatrix}
@@ -873,37 +886,38 @@ 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(H1, freqs, 'Hz'))), '-', 'DisplayName', '$|H_1|$');
-  plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
+  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');
+  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)]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/hinf_comp_filters.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/hinf_comp_filters.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:hinf_comp_filters
@@ -925,8 +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', '$1 + W_2 \Delta_2$');
@@ -942,11 +958,11 @@ The $\mathcal{H}_\infty$ synthesis thus allows to design filters such that the s
   set(gca, 'XTickLabel',[]);
   ylabel('Magnitude');
   ylim([1e-2, 1e1]);
-  legend('location', 'southeast');
+  legend('location', 'southeast', 'FontSize', 8);
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -959,12 +975,13 @@ 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
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/super_sensor_dynamical_uncertainty_Hinf.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/super_sensor_dynamical_uncertainty_Hinf.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:super_sensor_dynamical_uncertainty_Hinf
@@ -973,10 +990,8 @@ The $\mathcal{H}_\infty$ synthesis thus allows to design filters such that the s
 [[file:figs/super_sensor_dynamical_uncertainty_Hinf.png]]
 
 ** Super sensor noise
-We now compute the obtain Power Spectral Density of the super sensor's noise (Figure [[fig:psd_sensors_hinf_synthesis]]).
+We now compute the obtain Power Spectral Density of the super sensor's noise.
-
+The Amplitude Spectral Densities are shown in Figure [[fig:psd_sensors_hinf_synthesis]].
-The obtained RMS of the super sensor noise in the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ case are shown in Table [[tab:rms_noise_comp_H2_Hinf]].
-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.
 
 #+begin_src matlab
   PSD_S2   = abs(squeeze(freqresp(N2,    freqs, 'Hz'))).^2;
@@ -985,6 +1000,9 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
              abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
 #+end_src
 
+The obtained RMS of the super sensor noise in the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ case are shown in Table [[tab:rms_noise_comp_H2_Hinf]].
+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.
+
 #+begin_src matlab :exports none
   H2_filters = load('./mat/H2_filters.mat', 'H2', 'H1');
 
@@ -995,23 +1013,23 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, PSD_S1,   '-',   'DisplayName', '$\Phi_{n_1}$');
+  plot(freqs, sqrt(PSD_S1),   '-',   'DisplayName', '$\phi_{n_1}$');
-  plot(freqs, PSD_S2,   '-',   'DisplayName', '$\Phi_{n_2}$');
+  plot(freqs, sqrt(PSD_S2),   '-',   'DisplayName', '$\phi_{n_2}$');
-  plot(freqs, PSD_H2,   'k-',  'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$');
+  plot(freqs, sqrt(PSD_H2),   'k-',  'DisplayName', '$\phi_{n_{\mathcal{H}_2}}$');
-  plot(freqs, PSD_Hinf, 'k--', 'DisplayName', '$\Phi_{n_{\mathcal{H}_\infty}}$');
+  plot(freqs, sqrt(PSD_Hinf), 'k--', 'DisplayName', '$\phi_{n_{\mathcal{H}_\infty}}$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density $\left[ \frac{(m/s)^2}{Hz} \right]$');
+  xlabel('Frequency [Hz]'); ylabel('ASD $\left[ \frac{m/s}{\sqrt{Hz}} \right]$');
   hold off;
   xlim([freqs(1), freqs(end)]);
-  legend('location', 'northeast');
+  legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/psd_sensors_hinf_synthesis.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/psd_sensors_hinf_synthesis.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
 #+name: fig:psd_sensors_hinf_synthesis
-#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the
+#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
 #+RESULTS:
 [[file:figs/psd_sensors_hinf_synthesis.png]]
 
@@ -1048,7 +1066,7 @@ The sensor fusion architecture is shown in Figure [[fig:sensor_fusion_arch_full]
 
 #+name: fig:sensor_fusion_arch_full
 #+caption: Sensor fusion architecture with sensor dynamics uncertainty
-[[file:figs-tikz/sensor_fusion_arch_full.png]]
+[[file:figs-paper/sensor_fusion_arch_full.png]]
 
 The goal is to design complementary filters such that:
 - the maximum uncertainty of the super sensor is bounded to acceptable values (defined by $W_u(s)$)
@@ -1081,7 +1099,7 @@ The filter $H_2(s)$ is synthesized such that it:
 
 #+name: fig:mixed_h2_hinf_synthesis
 #+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
-[[file:figs-tikz/mixed_h2_hinf_synthesis.png]]
+[[file:figs-paper/mixed_h2_hinf_synthesis.png]]
 
 Let's see that
 with $H_1(s)= 1 - H_2(s)$
@@ -1120,38 +1138,38 @@ 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');
+  ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
-  set(gca, 'XTickLabel',[]);
   ylim([1e-3, 2]);
-  legend('location', 'southwest');
+  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)]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/htwo_hinf_comp_filters.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/htwo_hinf_comp_filters.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:htwo_hinf_comp_filters
@@ -1160,7 +1178,7 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt
 [[file:figs/htwo_hinf_comp_filters.png]]
 
 ** Obtained Super Sensor's noise
-The Power Spectral Density of the super sensor's noise is shown in Figure [[fig:psd_sensors_htwo_hinf_synthesis]].
+The Amplitude Spectral Density of the super sensor's noise is shown in Figure [[fig:psd_sensors_htwo_hinf_synthesis]].
 
 A time domain simulation is shown in Figure [[fig:super_sensor_time_domain_h2_hinf]].
 
@@ -1190,24 +1208,24 @@ The RMS values of the super sensor noise for the presented three synthesis are l
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, PSD_S1,     '-',   'DisplayName', '$\Phi_{n_1}$');
+  plot(freqs, sqrt(PSD_S1),     '-',   'DisplayName', '$\Phi_{n_1}$');
-  plot(freqs, PSD_S2,     '-',   'DisplayName', '$\Phi_{n_2}$');
+  plot(freqs, sqrt(PSD_S2),     '-',   'DisplayName', '$\Phi_{n_2}$');
-  plot(freqs, PSD_H2,     'k-',  'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$');
+  plot(freqs, sqrt(PSD_H2),     'k-',  'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$');
-  plot(freqs, PSD_Hinf,   'k--', 'DisplayName', '$\Phi_{n_{\mathcal{H}_\infty}}$');
+  plot(freqs, sqrt(PSD_Hinf),   'k--', 'DisplayName', '$\Phi_{n_{\mathcal{H}_\infty}}$');
-  plot(freqs, PSD_H2Hinf, 'k-.', 'DisplayName', '$\Phi_{n_{\mathcal{H}_2/\mathcal{H}_\infty}}$');
+  plot(freqs, sqrt(PSD_H2Hinf), 'k-.', 'DisplayName', '$\Phi_{n_{\mathcal{H}_2/\mathcal{H}_\infty}}$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$(m/s)^2/Hz$]');
+  xlabel('Frequency [Hz]'); ylabel('ASD $\left[ \frac{m/s}{\sqrt{Hz}} \right]$');
   hold off;
   xlim([freqs(1), freqs(end)]);
-  legend('location', 'northeast');
+  legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/psd_sensors_htwo_hinf_synthesis.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/psd_sensors_htwo_hinf_synthesis.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
 #+name: fig:psd_sensors_htwo_hinf_synthesis
-#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
+#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
 #+RESULTS:
 [[file:figs/psd_sensors_htwo_hinf_synthesis.png]]
 
@@ -1236,12 +1254,12 @@ The RMS values of the super sensor noise for the presented three synthesis are l
   plot(t, v, 'k--', 'DisplayName', '$x$');
   hold off;
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
-  legend('location', 'southwest');
+  legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3);
   ylim([-0.3, 0.3]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/super_sensor_time_domain_h2_hinf.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/super_sensor_time_domain_h2_hinf.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
 #+name: fig:super_sensor_time_domain_h2_hinf
@@ -1277,8 +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', '$1 + W_2 \Delta_2$');
@@ -1294,11 +1314,11 @@ The uncertainty on the super sensor's dynamics is shown in Figure [[fig:super_se
   set(gca, 'XTickLabel',[]);
   ylabel('Magnitude');
   ylim([1e-2, 1e1]);
-  legend('location', 'southeast');
+  legend('location', 'southeast', 'FontSize', 8);
   hold off;
 
   % Phase
-  ax2 = subplot(2,1,2);
+  ax2 = nexttile;
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
@@ -1307,16 +1327,16 @@ 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]); yticks(-180:90:180);
-  ylim([-180 180]);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
+
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf', 'width', 'full', 'height', 'full');
+  exportFig('figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf', 'width', 'half', 'height', 'tall');
 #+end_src
 
 #+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf

matlab/figs-journal

@@ -0,0 +1 @@
+../paper/figs

matlab/figs-tikz

@@ -1 +0,0 @@
-../tikz/figs