diff --git a/css/custom.css b/css/custom.css
deleted file mode 100644
index ff286c2..0000000
--- a/css/custom.css
+++ /dev/null
@@ -1,49 +0,0 @@
-.figure p{
- text-align: center;
-}
-
-.figure img{
- max-width:100%;
- display: block;
- margin: auto;
-}
-
-table {
- margin-left: auto;
- margin-right: auto;
-}
-
-.org-src-container > pre.src:before {
- display: inline;
- position: absolute;
- color: #808080;
- background-color: white;
- top: -10px;
- left: 10px;
- padding: 0px 4px;
- border: 1px solid #d0d0d0;
- font-size: 80%;
-}
-.org-src-container > pre {
- margin-top: 1.5em;
- position: relative;
- overflow: visible;
-}
-.org-src-container > pre > code.src:before {
- display: inline;
- position: absolute;
- color: #808080;
- background-color: white;
- top: -10px;
- left: 10px;
- padding: 0px 4px;
- border: 1px solid #d0d0d0;
- font-size: 80%;
-}
-.org-src-container > pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
-.org-src-container > pre.src-elisp:before { content: 'Emacs Lisp'; }
-.org-src-container > pre.src-sh:before { content: 'shell'; }
-.org-src-container > pre.src-bash:before { content: 'bash'; }
-.org-src-container > pre.src-org:before { content: 'Org mode'; }
-.org-src-container > pre.src-python:before { content: 'Python'; }
-.org-src-container > pre.src-matlab:before { content: 'Matlab'; }
diff --git a/css/htmlize.css b/css/htmlize.css
deleted file mode 100644
index 0b32a03..0000000
--- a/css/htmlize.css
+++ /dev/null
@@ -1,145 +0,0 @@
-.org-bold { /* bold */ font-weight: bold; }
-.org-bold-italic { /* bold-italic */ font-weight: bold; font-style: italic; }
-.org-buffer-menu-buffer { /* buffer-menu-buffer */ font-weight: bold; }
-.org-builtin { /* font-lock-builtin-face */ color: #7a378b; }
-.org-button { /* button */ text-decoration: underline; }
-.org-calendar-today { /* calendar-today */ text-decoration: underline; }
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-.org-change-log-conditionals { /* change-log-conditionals */ color: #a0522d; }
-.org-change-log-date { /* change-log-date */ color: #8b2252; }
-.org-change-log-email { /* change-log-email */ color: #a0522d; }
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diff --git a/css/readtheorg.css b/css/readtheorg.css
deleted file mode 100644
index 423a707..0000000
--- a/css/readtheorg.css
+++ /dev/null
@@ -1,1108 +0,0 @@
-@import url(https://fonts.googleapis.com/css?family=Lato:400,700,400italic,700italic|Roboto+Slab:400,700|Inconsolata:400,700);
-@import url(https://maxcdn.bootstrapcdn.com/font-awesome/4.3.0/css/font-awesome.min.css);
-
-h1,h2,h3,h4,h5,h6,legend{
- font-family:"Roboto Slab","ff-tisa-web-pro","Georgia",Arial,sans-serif;
- font-weight:700;
- margin-top:0;
-}
-
-h1{
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-.subtitle{
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-h3{
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-}
-
-h4{
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-}
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-h5{
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-}
-
-h6{
- font-size:100%;
-}
-
-h4,h5,h6{
- color:#2980B9;
- font-weight:300;
-}
-
-html{
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- -webkit-text-size-adjust:100%;
- font-size:100%;
- height:100%;
- overflow-x:hidden;
-}
-
-body{
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- color:#404040;
- font-family:"Lato","proxima-nova","Helvetica Neue",Arial,sans-serif;
- font-weight:normal;
- margin:0;
- min-height:100%;
- overflow-x:hidden;
-}
-
-#content{
- background:#fcfcfc;
- height:100%;
- margin-left:300px;
- /* margin:auto; */
- max-width:1200px;
- min-height:100%;
- padding:1.618em 3.236em;
-}
-
-p{
- font-size:16px;
- line-height:24px;
- margin:0px 0px 24px 0px;
-}
-
-b,strong{
- font-weight:bold}
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-blockquote{
- background-color: #F0F0F0;
- border-left:5px solid #CCCCCC;
- font-style:italic;
- line-height:24px;
- margin:0px 0px 24px 0px;
- /* margin-left:24px; */
- padding: 6px 20px;
-}
-
-ul,ol,dl{
- line-height:24px;
- list-style-image:none;
- /* list-style:none; */
- margin:0px 0px 24px 0px;
- padding:0;
-}
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-li{
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-}
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-dd{
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-}
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- line-height:24px;
- margin-bottom:24px}
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-#content .section ol li ul li,#content ol li ul li,article ol li ul li{
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-dl dt{
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-}
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-dl p,dl table,dl ul,dl ol{
- margin-bottom:12px !important;
-}
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-dl dd{
- margin:0 0 12px 24px;
-}
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-@media print{
- .codeblock,pre.src{
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-@media print{
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- *{
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- text-shadow:none !important;
- filter:none !important;
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- tr,img{
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- img{
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- @page{
- margin:0.5cm}
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- p,h2,h3{
- orphans:3;
- widows:3}
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- h2,h3{
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-}
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-@media print{
- #postamble{
- display:none}
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- #content{
- margin-left:0}
-}
-
-@media print{
- #table-of-contents{
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- @page{
- size: auto;
- margin: 25mm 25mm 25mm 25mm;}
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- body {
- margin: 0px;}
-}
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-@media screen and (max-width: 768px){
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-@media only screen and (max-width: 480px){
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-}
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-@media screen and (max-width: 768px){
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- #content #content{
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- #content.shift{
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- min-width:100%;
- left:85%;
- top:0;
- height:100%;
- overflow:hidden}
-}
-
-@media screen and (min-width: 1400px){
- #content{
- background:rgba(0,0,0,0.05)}
-
- #content{
- background:#fcfcfc}
-}
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-@media screen and (max-width: 768px){
- #copyright{
- width:85%;
- display:none}
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- #copyright.shift{
- display:block}
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- img{
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- height:auto}
-}
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-@media screen and (max-width: 480px){
- #content .sidebar{
- width:100%}
-}
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-code{
- background:#fff;
- border:solid 1px #e1e4e5;
- /* color:#000; for clickable code */
- font-family:Consolas,"Andale Mono WT","Andale Mono","Lucida Console","Lucida Sans Typewriter","DejaVu Sans Mono","Bitstream Vera Sans Mono","Liberation Mono","Nimbus Mono L",Monaco,"Courier New",Courier,monospace;
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- white-space:nowrap;
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-.codeblock-example:after{
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- background:#9B59B6;
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- padding:6px 12px}
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-.codeblock,pre.src,#content .literal-block{
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- padding:12px;
- overflow-x:auto;
- background:#fff;
- margin:1px 0 24px 0}
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-pre.src{
- /* color:#404040; */
- display:block;
- font-family:Consolas,"Andale Mono WT","Andale Mono","Lucida Console","Lucida Sans Typewriter","DejaVu Sans Mono","Bitstream Vera Sans Mono","Liberation Mono","Nimbus Mono L",Monaco,"Courier New",Courier,monospace;
- font-size:12px;
- line-height:1.5;
- margin:1px 0px 24px 0px;
- overflow:auto;
- padding:12px;
- white-space:pre;
-}
-
-.example{
- background:#f3f6f6;
- border:1px solid #e1e4e5;
- color:#404040;
- font-size: 12px;
- line-height: 1.5;
- margin-bottom:24px;
- padding:12px;
-}
-
-table{
- border-collapse:collapse;
- border-spacing:0;
- empty-cells:show;
- margin-bottom:24px;
- border-bottom:1px solid #e1e4e5;
-}
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-td{
- vertical-align:top}
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-table td,table th{
- font-size:90%;
- margin:0;
- overflow:visible;
- padding:8px 16px;
- background-color:white;
- border:1px solid #e1e4e5;
-}
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-table thead th{
- font-weight:bold;
- border-top:3px solid #e1e4e5;
- border-bottom:1px solid #e1e4e5;
-}
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-table caption{
- color:#000;
- font:italic 85%/1 arial,sans-serif;
- padding:1em 0;
-}
-
-table tr:nth-child(2n-1) td{
- background-color:#f3f6f6;
-}
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-table tr:nth-child(2n) td{
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-}
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-.figure p{
- color:#000;
- font:italic 85%/1 arial,sans-serif;
- padding:1em 0;
-}
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-.rotate-90{
- -webkit-transform:rotate(90deg);
- -moz-transform:rotate(90deg);
- -ms-transform:rotate(90deg);
- -o-transform:rotate(90deg);
- transform:rotate(90deg);
-}
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Sensor Fusion - Test Bench
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diff --git a/index.tex b/index.tex
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-% Created 2020-11-11 mer. 16:50
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- pdfkeywords={},
- pdfsubject={},
- pdfcreator={Emacs 27.1 (Org mode 9.4)},
- pdflang={English}}
-\begin{document}
-
-\maketitle
-\setcounter{tocdepth}{2}
-\tableofcontents
-
-
-In this document, we wish the experimentally validate sensor fusion of inertial sensors.
-
-This document is divided into the following sections:
-\begin{itemize}
-\item Section \ref{sec:experimental_setup}: the experimental setup is described
-\item Section \ref{sec:first_identification}: a first identification of the system dynamics is performed
-\item Section \ref{sec:integral_force_feedback}: the integral force feedback active damping technique is applied on the system
-\item Section \ref{sec:optimal_excitation}: the optimal excitation signal is determine in order to have the best possible system dynamics estimation
-\item Section \ref{sec:inertial_sensor_dynamics}: the inertial sensor dynamics are experimentally estimated
-\item Section \ref{sec:inertial_sensor_noise}: the inertial sensor noises are estimated and the \(\mathcal{H}_2\) synthesis of complementary filters is performed in order to yield a super sensor with minimal noise
-\item Section \ref{sec:inertial_sensor_uncertainty}: the dynamical uncertainty of the inertial sensors is estimated. Then the \(\mathcal{H}_\infty\) synthesis of complementary filters is performed in order to minimize the super sensor dynamical uncertainty
-\item Section \ref{sec:optimal_sensor_fusion}: Optimal sensor fusion is performed using the \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis
-\end{itemize}
-
-\section{Experimental Setup}
-\label{sec:org1f25ae6}
-\label{sec:experimental_setup}
-
-The goal of this experimental setup is to experimentally merge inertial sensors.
-To merge the sensors, optimal and robust complementary filters are designed.
-
-A schematic of the test-bench used is shown in Figure \ref{fig:exp_setup_sensor_fusion} and a picture of it is shown in Figure \ref{fig:test-bench-sensor-fusion-picture}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/exp_setup_sensor_fusion.png}
-\caption{\label{fig:exp_setup_sensor_fusion}Schematic of the test-bench}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/test-bench-sensor-fusion-picture.png}
-\caption{\label{fig:test-bench-sensor-fusion-picture}Picture of the test-bench}
-\end{figure}
-
-
-Two inertial sensors are used:
-\begin{itemize}
-\item An vertical accelerometer \emph{PCB 393B05} (\href{doc/TM-VIB-Seismic\_Lowres.pdf}{doc})
-\item A vertical geophone \emph{Mark Product L-22}
-\end{itemize}
-
-Basic characteristics of both sensors are shown in Tables \ref{tab:393B05_spec} and \ref{tab:L22_spec}.
-
-\begin{table}[htbp]
-\caption{\label{tab:393B05_spec}Accelerometer (393B05) Specifications}
-\centering
-\begin{tabular}{ll}
-\textbf{Specification} & \textbf{Value}\\
-\hline
-Sensitivity & 1.02 [V/(m/s2)]\\
-Resonant Frequency & > 2.5 [kHz]\\
-Resolution (1 to 10kHz) & 0.00004 [m/s2 rms]\\
-\end{tabular}
-\end{table}
-
-\begin{table}[htbp]
-\caption{\label{tab:L22_spec}Geophone (L22) Specifications}
-\centering
-\begin{tabular}{ll}
-\textbf{Specification} & \textbf{Value}\\
-\hline
-Sensitivity & To be measured [V/(m/s)]\\
-Resonant Frequency & 2 [Hz]\\
-\end{tabular}
-\end{table}
-
-The ADC used are the IO131 Speedgoat module (\href{https://www.speedgoat.com/products/io-connectivity-analog-io131}{link}) with a 16bit resolution over +/- 10V.
-
-The geophone signals are amplified using a DLPVA-100-B-D voltage amplified from Femto (\href{doc/de-dlpva-100-b.pdf}{doc}).
-The force sensor signal is amplified using a Low Noise Voltage Preamplifier from Ametek (\href{doc/model\_5113.pdf}{doc}).
-
-The excitation signal is amplified by a linear amplified from Cedrat (LA75B) with a gain equals to 20 (\href{doc/LA75B.pdf}{doc}).
-
-Geophone electronics:
-\begin{itemize}
-\item gain: 10 (20dB)
-\item low pass filter: 1.5Hz
-\item hifh pass filter: 100kHz (2nd order)
-\end{itemize}
-
-Force Sensor electronics:
-\begin{itemize}
-\item gain: 10 (20dB)
-\item low pass filter: 1st order at 3Hz
-\item high pass filter: 1st order at 30kHz
-\end{itemize}
-
-\section{First identification of the system}
-\label{sec:orgfd3d380}
-\label{sec:first_identification}
-In this section, a first identification of each elements of the system is performed.
-This include the dynamics from the actuator to the force sensor, interferometer and inertial sensors.
-
-Each of the dynamics is compared with the dynamics identified form a Simscape model.
-\subsection{Load Data}
-\label{sec:orge975c98}
-The data is loaded in the Matlab workspace.
-\begin{minted}[]{matlab}
-id_ol = load('identification_noise_bis.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-\end{minted}
-
-Then, any offset is removed.
-\begin{minted}[]{matlab}
-id_ol.d = detrend(id_ol.d, 0);
-id_ol.acc_1 = detrend(id_ol.acc_1, 0);
-id_ol.acc_2 = detrend(id_ol.acc_2, 0);
-id_ol.geo_1 = detrend(id_ol.geo_1, 0);
-id_ol.geo_2 = detrend(id_ol.geo_2, 0);
-id_ol.f_meas = detrend(id_ol.f_meas, 0);
-id_ol.u = detrend(id_ol.u, 0);
-\end{minted}
-
-\subsection{Excitation Signal}
-\label{sec:org5e8f3ce}
-The generated voltage used to excite the system is a white noise and can be seen in Figure \ref{fig:excitation_signal_first_identification}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/excitation_signal_first_identification.png}
-\caption{\label{fig:excitation_signal_first_identification}Voltage excitation signal}
-\end{figure}
-
-\subsection{Identified Plant}
-\label{sec:org326a4c0}
-The transfer function from the excitation voltage to the mass displacement and to the force sensor stack voltage are identified using the \texttt{tfestimate} command.
-
-\begin{minted}[]{matlab}
-Ts = id_ol.t(2) - id_ol.t(1);
-win = hann(ceil(10/Ts));
-\end{minted}
-
-\begin{minted}[]{matlab}
-[tf_fmeas_est, f] = tfestimate(id_ol.u, id_ol.f_meas, win, [], [], 1/Ts); % [V/V]
-[tf_G_ol_est, ~] = tfestimate(id_ol.u, id_ol.d, win, [], [], 1/Ts); % [m/V]
-\end{minted}
-
-The bode plots of the obtained dynamics are shown in Figures \ref{fig:force_sensor_bode_plot} and \ref{fig:displacement_sensor_bode_plot}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/force_sensor_bode_plot.png}
-\caption{\label{fig:force_sensor_bode_plot}Bode plot of the dynamics from excitation voltage to measured force sensor stack voltage}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/displacement_sensor_bode_plot.png}
-\caption{\label{fig:displacement_sensor_bode_plot}Bode plot of the dynamics from excitation voltage to displacement of the mass as measured by the interferometer}
-\end{figure}
-
-\subsection{Simscape Model - Comparison}
-\label{sec:org8797fe4}
-A simscape model representing the test-bench has been developed.
-The same transfer functions as the one identified using the test-bench can be obtained thanks to the simscape model.
-
-They are compared in Figure \ref{fig:simscape_comp_iff_plant} and \ref{fig:simscape_comp_disp_plant}.
-It is shown that there is a good agreement between the model and the experiment.
-
-\begin{minted}[]{matlab}
-load('piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/simscape_comp_iff_plant.png}
-\caption{\label{fig:simscape_comp_iff_plant}Comparison of the dynamics from excitation voltage to measured force sensor stack voltage - Identified dynamics and Simscape Model}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/simscape_comp_disp_plant.png}
-\caption{\label{fig:simscape_comp_disp_plant}Comparison of the dynamics from excitation voltage to measured mass displacement - Identified dynamics and Simscape Model}
-\end{figure}
-
-\subsection{Integral Force Feedback}
-\label{sec:org985543a}
-The force sensor stack can be used to damp the system.
-This makes the system easier to excite properly without too much amplification near resonances.
-
-This is done thanks to the integral force feedback control architecture.
-
-The force sensor stack signal is integrated (or rather low pass filtered) and fed back to the force sensor stacks.
-
-The low pass filter used as the controller is defined below:
-\begin{minted}[]{matlab}
-Kiff = 102/(s + 2*pi*2);
-\end{minted}
-
-The integral force feedback control strategy is applied to the simscape model as well as to the real test bench.
-
-The damped system is then identified again using a noise excitation.
-
-The data is loaded into Matlab and any offset is removed.
-\begin{minted}[]{matlab}
-id_cl = load('identification_noise_iff_bis.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-
-id_cl.d = detrend(id_cl.d, 0);
-id_cl.acc_1 = detrend(id_cl.acc_1, 0);
-id_cl.acc_2 = detrend(id_cl.acc_2, 0);
-id_cl.geo_1 = detrend(id_cl.geo_1, 0);
-id_cl.geo_2 = detrend(id_cl.geo_2, 0);
-id_cl.f_meas = detrend(id_cl.f_meas, 0);
-id_cl.u = detrend(id_cl.u, 0);
-\end{minted}
-
-The transfer functions are estimated using \texttt{tfestimate}.
-\begin{minted}[]{matlab}
-[tf_G_cl_est, ~] = tfestimate(id_cl.u, id_cl.d, win, [], [], 1/Ts);
-[co_G_cl_est, ~] = mscohere( id_cl.u, id_cl.d, win, [], [], 1/Ts);
-\end{minted}
-
-The dynamics from driving voltage to the measured displacement are compared both in the open-loop and IFF case, and for the test-bench experimental identification and for the Simscape model in Figure \ref{fig:iff_ol_cl_identified_simscape_comp}.
-This shows that the Integral Force Feedback architecture effectively damps the first resonance of the system.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/iff_ol_cl_identified_simscape_comp.png}
-\caption{\label{fig:iff_ol_cl_identified_simscape_comp}Comparison of the open-loop and closed-loop (IFF) dynamics for both the real identification and the Simscape one}
-\end{figure}
-
-\subsection{Inertial Sensors}
-\label{sec:orgcdb3347}
-In order to estimate the dynamics of the inertial sensor (the transfer function from the ``absolute'' displacement to the measured voltage), the following experiment can be performed:
-\begin{itemize}
-\item The mass is excited such that is relative displacement as measured by the interferometer is much larger that the ground ``absolute'' motion.
-\item The transfer function from the measured displacement by the interferometer to the measured voltage generated by the inertial sensors can be estimated.
-\end{itemize}
-
-The first point is quite important in order to have a good coherence between the interferometer measurement and the inertial sensor measurement.
-
-Here, a first identification is performed were the excitation signal is a white noise.
-
-
-As usual, the data is loaded and any offset is removed.
-\begin{minted}[]{matlab}
-id = load('identification_noise_opt_iff.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-
-id.d = detrend(id.d, 0);
-id.acc_1 = detrend(id.acc_1, 0);
-id.acc_2 = detrend(id.acc_2, 0);
-id.geo_1 = detrend(id.geo_1, 0);
-id.geo_2 = detrend(id.geo_2, 0);
-id.f_meas = detrend(id.f_meas, 0);
-\end{minted}
-
-Then the transfer functions from the measured displacement by the interferometer to the generated voltage of the inertial sensors are computed..
-\begin{minted}[]{matlab}
-Ts = id.t(2) - id.t(1);
-win = hann(ceil(10/Ts));
-\end{minted}
-
-\begin{minted}[]{matlab}
-[tf_acc1_est, f] = tfestimate(id.d, id.acc_1, win, [], [], 1/Ts);
-[co_acc1_est, ~] = mscohere( id.d, id.acc_1, win, [], [], 1/Ts);
-[tf_acc2_est, ~] = tfestimate(id.d, id.acc_2, win, [], [], 1/Ts);
-[co_acc2_est, ~] = mscohere( id.d, id.acc_2, win, [], [], 1/Ts);
-
-[tf_geo1_est, ~] = tfestimate(id.d, id.geo_1, win, [], [], 1/Ts);
-[co_geo1_est, ~] = mscohere( id.d, id.geo_1, win, [], [], 1/Ts);
-[tf_geo2_est, ~] = tfestimate(id.d, id.geo_2, win, [], [], 1/Ts);
-[co_geo2_est, ~] = mscohere( id.d, id.geo_2, win, [], [], 1/Ts);
-\end{minted}
-
-The same transfer functions are estimated using the Simscape model.
-
-The obtained dynamics of the accelerometer are compared in Figure \ref{fig:comp_dynamics_accelerometer} while the one of the geophones are compared in Figure \ref{fig:comp_dynamics_geophone}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/comp_dynamics_accelerometer.png}
-\caption{\label{fig:comp_dynamics_accelerometer}Comparison of the measured accelerometer dynamics}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/comp_dynamics_geophone.png}
-\caption{\label{fig:comp_dynamics_geophone}Comparison of the measured geophone dynamics}
-\end{figure}
-
-\section{Optimal IFF Development}
-\label{sec:org2ad9f9e}
-\label{sec:integral_force_feedback}
-In this section, a proper identification of the transfer function from the force actuator to the force sensor is performed.
-Then, an optimal IFF controller is developed and applied experimentally.
-
-The damped system is identified to verified the effectiveness of the added method.
-\subsection{Load Data}
-\label{sec:org648d21b}
-The experimental data is loaded and any offset is removed.
-\begin{minted}[]{matlab}
-id_ol = load('identification_noise_bis.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-\end{minted}
-
-\begin{minted}[]{matlab}
-id_ol.d = detrend(id_ol.d, 0);
-id_ol.acc_1 = detrend(id_ol.acc_1, 0);
-id_ol.acc_2 = detrend(id_ol.acc_2, 0);
-id_ol.geo_1 = detrend(id_ol.geo_1, 0);
-id_ol.geo_2 = detrend(id_ol.geo_2, 0);
-id_ol.f_meas = detrend(id_ol.f_meas, 0);
-id_ol.u = detrend(id_ol.u, 0);
-\end{minted}
-
-\subsection{Experimental Data}
-\label{sec:org4151558}
-The transfer function from force actuator to force sensors is estimated.
-
-The coherence shown in Figure \ref{fig:iff_identification_coh} shows that the excitation signal is good enough.
-
-\begin{minted}[]{matlab}
-Ts = id_ol.t(2) - id_ol.t(1);
-win = hann(ceil(10/Ts));
-\end{minted}
-
-\begin{minted}[]{matlab}
-[tf_fmeas_est, f] = tfestimate(id_ol.u, id_ol.f_meas, win, [], [], 1/Ts); % [V/m]
-[co_fmeas_est, ~] = mscohere( id_ol.u, id_ol.f_meas, win, [], [], 1/Ts);
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/iff_identification_coh.png}
-\caption{\label{fig:iff_identification_coh}Coherence for the identification of the IFF plant}
-\end{figure}
-
-The obtained dynamics is shown in Figure \ref{fig:iff_identification_bode_plot}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/iff_identification_bode_plot.png}
-\caption{\label{fig:iff_identification_bode_plot}Bode plot of the identified IFF plant}
-\end{figure}
-
-\subsection{Model of the IFF Plant}
-\label{sec:org50bd8ad}
-In order to plot the root locus for the IFF control strategy, a model of the identified plant is developed.
-
-It consists of several poles and zeros are shown below.
-\begin{minted}[]{matlab}
-wz = 2*pi*102;
-xi_z = 0.01;
-wp = 2*pi*239.4;
-xi_p = 0.015;
-
-Giff = 2.2*(s^2 + 2*xi_z*s*wz + wz^2)/(s^2 + 2*xi_p*s*wp + wp^2) * ... % Dynamics
- 10*(s/3/pi/(1 + s/3/pi)) * ... % Low pass filter and gain of the voltage amplifier
- exp(-Ts*s); % Time delay induced by ADC/DAC
-\end{minted}
-
-The comparison of the identified dynamics and the developed model is done in Figure \ref{fig:iff_plant_model}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/iff_plant_model.png}
-\caption{\label{fig:iff_plant_model}IFF Plant + Model}
-\end{figure}
-
-\subsection{Root Locus and optimal Controller}
-\label{sec:orgd2946d4}
-Now, the root locus for the Integral Force Feedback strategy is computed and shown in Figure \ref{fig:iff_root_locus}.
-
-Note that the controller used is not a pure integrator but rather a first order low pass filter with a cut-off frequency set at 2Hz.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/iff_root_locus.png}
-\caption{\label{fig:iff_root_locus}Root Locus for the IFF control}
-\end{figure}
-
-The controller that yield maximum damping (shown by the red cross in Figure \ref{fig:iff_root_locus}) is:
-\begin{minted}[]{matlab}
-Kiff_opt = 102/(s + 2*pi*2);
-\end{minted}
-
-\subsection{Verification of the achievable damping}
-\label{sec:org163da4b}
-A new identification is performed with the IFF control strategy applied to the system.
-
-Data is loaded and offset removed.
-\begin{minted}[]{matlab}
-id_cl = load('identification_noise_iff_bis.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-\end{minted}
-
-\begin{minted}[]{matlab}
-id_cl.d = detrend(id_cl.d, 0);
-id_cl.acc_1 = detrend(id_cl.acc_1, 0);
-id_cl.acc_2 = detrend(id_cl.acc_2, 0);
-id_cl.geo_1 = detrend(id_cl.geo_1, 0);
-id_cl.geo_2 = detrend(id_cl.geo_2, 0);
-id_cl.f_meas = detrend(id_cl.f_meas, 0);
-id_cl.u = detrend(id_cl.u, 0);
-\end{minted}
-
-The open-loop and closed-loop dynamics are estimated.
-\begin{minted}[]{matlab}
-[tf_G_ol_est, f] = tfestimate(id_ol.u, id_ol.d, win, [], [], 1/Ts);
-[co_G_ol_est, ~] = mscohere( id_ol.u, id_ol.d, win, [], [], 1/Ts);
-[tf_G_cl_est, ~] = tfestimate(id_cl.u, id_cl.d, win, [], [], 1/Ts);
-[co_G_cl_est, ~] = mscohere( id_cl.u, id_cl.d, win, [], [], 1/Ts);
-\end{minted}
-
-The obtained coherence is shown in Figure \ref{fig:Gd_identification_iff_coherence} and the dynamics in Figure \ref{fig:Gd_identification_iff_bode_plot}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/Gd_identification_iff_coherence.png}
-\caption{\label{fig:Gd_identification_iff_coherence}Coherence for the transfer function from F to d, with and without IFF}
-\end{figure}
-
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/Gd_identification_iff_bode_plot.png}
-\caption{\label{fig:Gd_identification_iff_bode_plot}Coherence for the transfer function from F to d, with and without IFF}
-\end{figure}
-
-\section{Generate the excitation signal}
-\label{sec:orgad4b447}
-\label{sec:optimal_excitation}
-In order to properly estimate the dynamics of the inertial sensor, the excitation signal must be properly chosen.
-
-The requirements on the excitation signal is:
-\begin{itemize}
-\item General much larger motion that the measured motion during the huddle test
-\item Don't damage the actuator
-\end{itemize}
-
-To determine the perfect voltage signal to be generated, we need two things:
-\begin{itemize}
-\item the transfer function from voltage to mass displacement
-\item the PSD of the measured motion by the inertial sensors
-\item not saturate the sensor signals
-\item provide enough signal/noise ratio (good coherence) in the frequency band of interest (\textasciitilde{}0.5Hz to 3kHz)
-\end{itemize}
-\subsection{Transfer function from excitation signal to displacement}
-\label{sec:orgeb37755}
-Let's first estimate the transfer function from the excitation signal in [V] to the generated displacement in [m] as measured by the inteferometer.
-
-\begin{minted}[]{matlab}
-id_cl = load('identification_noise_iff_bis.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-\end{minted}
-
-\begin{minted}[]{matlab}
-Ts = id_cl.t(2) - id_cl.t(1);
-win = hann(ceil(10/Ts));
-\end{minted}
-
-\begin{minted}[]{matlab}
-[tf_G_cl_est, f] = tfestimate(id_cl.u, id_cl.d, win, [], [], 1/Ts);
-[co_G_cl_est, ~] = mscohere( id_cl.u, id_cl.d, win, [], [], 1/Ts);
-\end{minted}
-
-Approximate transfer function from voltage output to generated displacement when IFF is used, in [m/V].
-\begin{minted}[]{matlab}
-G_d_est = -5e-6*(2*pi*230)^2/(s^2 + 2*0.3*2*pi*240*s + (2*pi*240)^2);
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/Gd_plant_estimation.png}
-\caption{\label{fig:Gd_plant_estimation}Estimation of the transfer function from the excitation signal to the generated displacement}
-\end{figure}
-
-\subsection{Motion measured during Huddle test}
-\label{sec:org2632bf6}
-We now compute the PSD of the measured motion by the inertial sensors during the huddle test.
-\begin{minted}[]{matlab}
-ht = load('huddle_test.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-ht.d = detrend(ht.d, 0);
-ht.acc_1 = detrend(ht.acc_1, 0);
-ht.acc_2 = detrend(ht.acc_2, 0);
-ht.geo_1 = detrend(ht.geo_1, 0);
-ht.geo_2 = detrend(ht.geo_2, 0);
-\end{minted}
-
-\begin{minted}[]{matlab}
-[p_d, f] = pwelch(ht.d, win, [], [], 1/Ts);
-[p_acc1, ~] = pwelch(ht.acc_1, win, [], [], 1/Ts);
-[p_acc2, ~] = pwelch(ht.acc_2, win, [], [], 1/Ts);
-[p_geo1, ~] = pwelch(ht.geo_1, win, [], [], 1/Ts);
-[p_geo2, ~] = pwelch(ht.geo_2, win, [], [], 1/Ts);
-\end{minted}
-
-Using an estimated model of the sensor dynamics from the documentation of the sensors, we can compute the ASD of the motion in \(m/\sqrt{Hz}\) measured by the sensors.
-\begin{minted}[]{matlab}
-G_acc = 1/(1 + s/2/pi/2500); % [V/(m/s2)]
-G_geo = -120*s^2/(s^2 + 2*0.7*2*pi*2*s + (2*pi*2)^2); % [V/(m/s)]
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/huddle_test_psd_motion.png}
-\caption{\label{fig:huddle_test_psd_motion}ASD of the motion measured by the sensors}
-\end{figure}
-
-From the ASD of the motion measured by the sensors, we can create an excitation signal that will generate much motion motion that the motion under no excitation.
-
-We create \texttt{G\_exc} that corresponds to the wanted generated motion.
-\begin{minted}[]{matlab}
-G_exc = 0.2e-6/(1 + s/2/pi/2)/(1 + s/2/pi/50);
-\end{minted}
-
-And we create a time domain signal \texttt{y\_d} that have the spectral density described by \texttt{G\_exc}.
-\begin{minted}[]{matlab}
-Fs = 1/Ts;
-t = 0:Ts:180; % Time Vector [s]
-u = sqrt(Fs/2)*randn(length(t), 1); % Signal with an ASD equal to one
-
-y_d = lsim(G_exc, u, t);
-\end{minted}
-
-\begin{minted}[]{matlab}
-[pxx, ~] = pwelch(y_d, win, 0, [], Fs);
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/comp_huddle_test_excited_motion_psd.png}
-\caption{\label{fig:comp_huddle_test_excited_motion_psd}Comparison of the ASD of the motion during Huddle and the wanted generated motion}
-\end{figure}
-
-
-We can now generate the voltage signal that will generate the wanted motion.
-\begin{minted}[]{matlab}
-y_v = lsim(G_exc * ... % from unit PSD to shaped PSD
- (1 + s/2/pi/50) * ... % Inverse of pre-filter included in the Simulink file
- 1/G_d_est * ... % Wanted displacement => required voltage
- 1/(1 + s/2/pi/5e3), ... % Add some high frequency filtering
- u, t);
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/optimal_exc_signal_time.png}
-\caption{\label{fig:optimal_exc_signal_time}Generated excitation signal}
-\end{figure}
-
-\section{Identification of the Inertial Sensors Dynamics}
-\label{sec:org15a1119}
-\label{sec:inertial_sensor_dynamics}
-Using the excitation signal generated in Section \ref{sec:optimal_excitation}, the dynamics of the inertial sensors are identified.
-\subsection{Load Data}
-\label{sec:org8165027}
-Both the measurement data during the identification test and during an ``huddle test'' are loaded.
-\begin{minted}[]{matlab}
-id = load('identification_noise_opt_iff.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-ht = load('huddle_test.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-\end{minted}
-
-\begin{minted}[]{matlab}
-ht.d = detrend(ht.d, 0);
-ht.acc_1 = detrend(ht.acc_1, 0);
-ht.acc_2 = detrend(ht.acc_2, 0);
-ht.geo_1 = detrend(ht.geo_1, 0);
-ht.geo_2 = detrend(ht.geo_2, 0);
-ht.f_meas = detrend(ht.f_meas, 0);
-\end{minted}
-
-\begin{minted}[]{matlab}
-id.d = detrend(id.d, 0);
-id.acc_1 = detrend(id.acc_1, 0);
-id.acc_2 = detrend(id.acc_2, 0);
-id.geo_1 = detrend(id.geo_1, 0);
-id.geo_2 = detrend(id.geo_2, 0);
-id.f_meas = detrend(id.f_meas, 0);
-\end{minted}
-
-\subsection{Compare PSD during Huddle and and during identification}
-\label{sec:org6900214}
-The Power Spectral Density of the measured motion during the huddle test and during the identification test are compared in Figures \ref{fig:comp_psd_huddle_test_identification_acc} and \ref{fig:comp_psd_huddle_test_identification_geo}.
-
-\begin{minted}[]{matlab}
-Ts = ht.t(2) - ht.t(1);
-win = hann(ceil(10/Ts));
-\end{minted}
-
-\begin{minted}[]{matlab}
-[p_id_d, f] = pwelch(id.d, win, [], [], 1/Ts);
-[p_id_acc1, ~] = pwelch(id.acc_1, win, [], [], 1/Ts);
-[p_id_acc2, ~] = pwelch(id.acc_2, win, [], [], 1/Ts);
-[p_id_geo1, ~] = pwelch(id.geo_1, win, [], [], 1/Ts);
-[p_id_geo2, ~] = pwelch(id.geo_2, win, [], [], 1/Ts);
-\end{minted}
-
-\begin{minted}[]{matlab}
-[p_ht_d, ~] = pwelch(ht.d, win, [], [], 1/Ts);
-[p_ht_acc1, ~] = pwelch(ht.acc_1, win, [], [], 1/Ts);
-[p_ht_acc2, ~] = pwelch(ht.acc_2, win, [], [], 1/Ts);
-[p_ht_geo1, ~] = pwelch(ht.geo_1, win, [], [], 1/Ts);
-[p_ht_geo2, ~] = pwelch(ht.geo_2, win, [], [], 1/Ts);
-[p_ht_fmeas, ~] = pwelch(ht.f_meas, win, [], [], 1/Ts);
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/comp_psd_huddle_test_identification_acc.png}
-\caption{\label{fig:comp_psd_huddle_test_identification_acc}Comparison of the PSD of the measured motion during the Huddle test and during the identification}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/comp_psd_huddle_test_identification_geo.png}
-\caption{\label{fig:comp_psd_huddle_test_identification_geo}Comparison of the PSD of the measured motion during the Huddle test and during the identification}
-\end{figure}
-
-\subsection{Compute transfer functions}
-\label{sec:orgb24ac36}
-The transfer functions from the motion as measured by the interferometer (and that should represent the absolute motion of the mass) to the inertial sensors are estimated:
-\begin{minted}[]{matlab}
-[tf_acc1_est, f] = tfestimate(id.d, id.acc_1, win, [], [], 1/Ts);
-[co_acc1_est, ~] = mscohere( id.d, id.acc_1, win, [], [], 1/Ts);
-[tf_acc2_est, ~] = tfestimate(id.d, id.acc_2, win, [], [], 1/Ts);
-[co_acc2_est, ~] = mscohere( id.d, id.acc_2, win, [], [], 1/Ts);
-
-[tf_geo1_est, ~] = tfestimate(id.d, id.geo_1, win, [], [], 1/Ts);
-[co_geo1_est, ~] = mscohere( id.d, id.geo_1, win, [], [], 1/Ts);
-[tf_geo2_est, ~] = tfestimate(id.d, id.geo_2, win, [], [], 1/Ts);
-[co_geo2_est, ~] = mscohere( id.d, id.geo_2, win, [], [], 1/Ts);
-\end{minted}
-
-The obtained coherence are shown in Figure \ref{fig:id_sensor_dynamics_coherence}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/id_sensor_dynamics_coherence.png}
-\caption{\label{fig:id_sensor_dynamics_coherence}Coherence for the estimation of the sensor dynamics}
-\end{figure}
-
-We also make a simplified model of the inertial sensors to be compared with the identified dynamics.
-\begin{minted}[]{matlab}
-G_acc = 1/(1 + s/2/pi/2500); % [V/(m/s2)]
-G_geo = -1200*s^2/(s^2 + 2*0.7*2*pi*2*s + (2*pi*2)^2); % [[V/(m/s)]
-\end{minted}
-
-The model and identified dynamics show good agreement (Figures \ref{fig:id_sensor_dynamics_accelerometers} and \ref{fig:id_sensor_dynamics_geophones}.)
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/id_sensor_dynamics_accelerometers.png}
-\caption{\label{fig:id_sensor_dynamics_accelerometers}Identified dynamics of the accelerometers}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/id_sensor_dynamics_geophones.png}
-\caption{\label{fig:id_sensor_dynamics_geophones}Identified dynamics of the geophones}
-\end{figure}
-
-\section{Inertial Sensor Noise and the \(\mathcal{H}_2\) Synthesis of complementary filters}
-\label{sec:orgba1dbbc}
-\label{sec:inertial_sensor_noise}
-In this section, the noise of the inertial sensors (geophones and accelerometers) is estimated.
-\subsection{Load Data}
-\label{sec:orga473e53}
-As before, the identification data is loaded and any offset if removed.
-\begin{minted}[]{matlab}
-id = load('identification_noise_opt_iff.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-\end{minted}
-
-\begin{minted}[]{matlab}
-id.d = detrend(id.d, 0);
-id.acc_1 = detrend(id.acc_1, 0);
-id.acc_2 = detrend(id.acc_2, 0);
-id.geo_1 = detrend(id.geo_1, 0);
-id.geo_2 = detrend(id.geo_2, 0);
-id.f_meas = detrend(id.f_meas, 0);
-\end{minted}
-
-\subsection{ASD of the Measured displacement}
-\label{sec:orga1e0863}
-The Power Spectral Density of the displacement as measured by the interferometer and the inertial sensors is computed.
-\begin{minted}[]{matlab}
-Ts = id.t(2) - id.t(1);
-win = hann(ceil(10/Ts));
-\end{minted}
-
-\begin{minted}[]{matlab}
-[p_id_d, f] = pwelch(id.d, win, [], [], 1/Ts);
-[p_id_acc1, ~] = pwelch(id.acc_1, win, [], [], 1/Ts);
-[p_id_acc2, ~] = pwelch(id.acc_2, win, [], [], 1/Ts);
-[p_id_geo1, ~] = pwelch(id.geo_1, win, [], [], 1/Ts);
-[p_id_geo2, ~] = pwelch(id.geo_2, win, [], [], 1/Ts);
-\end{minted}
-
-Let's use a model of the accelerometer and geophone to compute the motion from the measured voltage.
-\begin{minted}[]{matlab}
-G_acc = 1/(1 + s/2/pi/2500); % [V/(m/s2)]
-G_geo = -1200*s^2/(s^2 + 2*0.7*2*pi*2*s + (2*pi*2)^2); % [[V/(m/s)]
-\end{minted}
-
-The obtained ASD in \(m/\sqrt{Hz}\) is shown in Figure \ref{fig:measure_displacement_all_sensors}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/measure_displacement_all_sensors.png}
-\caption{\label{fig:measure_displacement_all_sensors}ASD of the measured displacement as measured by all the sensors}
-\end{figure}
-
-\subsection{ASD of the Sensor Noise}
-\label{sec:org642e324}
-The noise of a sensor can be estimated using two identical sensors by computing:
-\begin{itemize}
-\item the Power Spectral Density of the measured motion by the two sensors
-\item the Cross Spectral Density between the two sensors (coherence)
-\end{itemize}
-
-This technique to estimate the sensor noise is described in \cite{barzilai98_techn_measur_noise_sensor_presen}.
-
-The Power Spectral Density of the sensor noise can be estimated using the following equation:
-\begin{equation}
- |S_n(\omega)| = |S_{x_1}(\omega)| \Big( 1 - \gamma_{x_1 x_2}(\omega) \Big)
-\end{equation}
-with \(S_{x_1}\) the PSD of one of the sensor and \(\gamma_{x_1 x_2}\) the coherence between the two sensors.
-
-The coherence between the two accelerometers and between the two geophones is computed.
-\begin{minted}[]{matlab}
-[coh_acc, ~] = mscohere(id.acc_1, id.acc_2, win, [], [], 1/Ts);
-[coh_geo, ~] = mscohere(id.geo_1, id.geo_2, win, [], [], 1/Ts);
-\end{minted}
-
-Finally, the Power Spectral Density of the sensors is computed and converted in \([m^2/Hz]\).
-\begin{minted}[]{matlab}
-pN_acc = p_id_acc1.*(1 - coh_acc) .* ... % [V^2/Hz]
- 1./abs(squeeze(freqresp(G_acc*s^2, f, 'Hz'))).^2; % [(m/V)^2]
-pN_geo = p_id_geo1.*(1 - coh_geo) .* ... % [V^2/Hz]
- 1./abs(squeeze(freqresp(G_geo*s, f, 'Hz'))).^2; % [(m/V)^2]
-\end{minted}
-
-The ASD of obtained noises are compared with the ASD of the measured signals in Figure \ref{fig:noise_inertial_sensors_comparison}.
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/noise_inertial_sensors_comparison.png}
-\caption{\label{fig:noise_inertial_sensors_comparison}Comparison of the computed ASD of the noise of the two inertial sensors}
-\end{figure}
-
-\subsection{Noise Model}
-\label{sec:org162c1bf}
-Transfer functions are adjusted in order to fit the ASD of the sensor noises (expressed in \([m/s/\sqrt{Hz}]\) for more easy fitting).
-
-These transfer functions are defined below and compared with the measured ASD in Figure \ref{fig:noise_models_velocity}.
-\begin{minted}[]{matlab}
-N_acc = 1*(s/(2*pi*2000) + 1)^2/(s + 0.1*2*pi)/(s + 1e3*2*pi); % [m/sqrt(Hz)]
-N_geo = 4e-4*(s/(2*pi*200) + 1)/(s + 1e3*2*pi); % [m/sqrt(Hz)]
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/noise_models_velocity.png}
-\caption{\label{fig:noise_models_velocity}ASD of the velocity noise measured by the sensors and the noise models}
-\end{figure}
-
-\subsection{\(\mathcal{H}_2\) Synthesis of the Complementary Filters}
-\label{sec:org2d65000}
-We now wish to synthesize two complementary filters to merge the geophone and the accelerometer signal in such a way that the fused signal has the lowest possible RMS noise.
-
-To do so, we use the \(\mathcal{H}_2\) synthesis where the transfer functions representing the noise density of both sensors are used as weights.
-
-The generalized plant used for the synthesis is defined below.
-\begin{minted}[]{matlab}
-P = [0 N_acc 1;
- N_geo -N_acc 0];
-\end{minted}
-
-And the \(\mathcal{H}_2\) synthesis is done using the \texttt{h2syn} command.
-\begin{minted}[]{matlab}
-[H_geo, ~, gamma] = h2syn(P, 1, 1);
-H_acc = 1 - H_geo;
-\end{minted}
-
-The obtained complementary filters are shown in Figure \ref{fig:complementary_filters_velocity_H2}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/complementary_filters_velocity_H2.png}
-\caption{\label{fig:complementary_filters_velocity_H2}Obtained Complementary Filters}
-\end{figure}
-
-\subsection{Results}
-\label{sec:orgef78a56}
-Finally, the signals of both sensors are merged using the complementary filters and the super sensor noise is estimated and compared with the individual sensor noises in Figure \ref{fig:super_sensor_noise_asd_velocity}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/super_sensor_noise_asd_velocity.png}
-\caption{\label{fig:super_sensor_noise_asd_velocity}ASD of the super sensor noise (velocity)}
-\end{figure}
-
-Finally, the Cumulative Power Spectrum is computed and compared in Figure \ref{fig:super_sensor_noise_cas_velocity}.
-\begin{minted}[]{matlab}
-[~, i_1Hz] = min(abs(f - 1));
-\end{minted}
-
-\begin{minted}[]{matlab}
-CPS_acc = 1/pi*flip(-cumtrapz(2*pi*flip(f), flip((pN_acc.*(2*pi*f)).^2)));
-CPS_geo = 1/pi*flip(-cumtrapz(2*pi*flip(f), flip((pN_geo.*(2*pi*f)).^2)));
-CPS_SS = 1/pi*flip(-cumtrapz(2*pi*flip(f), flip((pN_acc.*(2*pi*f)).^2.*abs(squeeze(freqresp(H_acc, f, 'Hz'))).^2 + (pN_geo.*(2*pi*f)).^2.*abs(squeeze(freqresp(H_geo, f, 'Hz'))).^2)));
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/super_sensor_noise_cas_velocity.png}
-\caption{\label{fig:super_sensor_noise_cas_velocity}Cumulative Power Spectrum of the Sensor Noise (velocity)}
-\end{figure}
-
-\section{Inertial Sensor Dynamics Uncertainty and the \(\mathcal{H}_\infty\) Synthesis of complementary filters}
-\label{sec:org4c06165}
-\label{sec:inertial_sensor_uncertainty}
-When merging two sensors, it is important to be sure that we correctly know the sensor dynamics near the merging frequency.
-Thus, identifying the uncertainty on the sensor dynamics is quite important to perform a robust merging.
-\subsection{Load Data}
-\label{sec:orgdc287f4}
-Data is loaded and offset is removed.
-
-\begin{minted}[]{matlab}
-id = load('identification_noise_opt_iff.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
-\end{minted}
-
-\begin{minted}[]{matlab}
-id.d = detrend(id.d, 0);
-id.acc_1 = detrend(id.acc_1, 0);
-id.acc_2 = detrend(id.acc_2, 0);
-id.geo_1 = detrend(id.geo_1, 0);
-id.geo_2 = detrend(id.geo_2, 0);
-id.f_meas = detrend(id.f_meas, 0);
-\end{minted}
-
-\subsection{Compute the dynamics of both sensors}
-\label{sec:org000ee7e}
-The dynamics of inertial sensors are estimated (in \([V/m]\)).
-\begin{minted}[]{matlab}
-Ts = id.t(2) - id.t(1);
-win = hann(ceil(10/Ts));
-\end{minted}
-
-\begin{minted}[]{matlab}
-[tf_acc1_est, f] = tfestimate(id.d, id.acc_1, win, [], [], 1/Ts);
-[co_acc1_est, ~] = mscohere( id.d, id.acc_1, win, [], [], 1/Ts);
-[tf_acc2_est, ~] = tfestimate(id.d, id.acc_2, win, [], [], 1/Ts);
-[co_acc2_est, ~] = mscohere( id.d, id.acc_2, win, [], [], 1/Ts);
-
-[tf_geo1_est, ~] = tfestimate(id.d, id.geo_1, win, [], [], 1/Ts);
-[co_geo1_est, ~] = mscohere( id.d, id.geo_1, win, [], [], 1/Ts);
-[tf_geo2_est, ~] = tfestimate(id.d, id.geo_2, win, [], [], 1/Ts);
-[co_geo2_est, ~] = mscohere( id.d, id.geo_2, win, [], [], 1/Ts);
-\end{minted}
-
-The (nominal) models of the inertial sensors from the absolute displacement to the generated voltage are defined below:
-\begin{minted}[]{matlab}
-G_acc = 1/(1 + s/2/pi/2000)
-G_geo = -1200*s^2/(s^2 + 2*0.7*2*pi*2*s + (2*pi*2)^2);
-\end{minted}
-
-These models are very simplistic models, and we then take into account the un-modelled dynamics with dynamical uncertainty.
-
-\subsection{Dynamics uncertainty estimation}
-\label{sec:org3425338}
-Weights representing the dynamical uncertainty of the sensors are defined below.
-\begin{minted}[]{matlab}
-w_acc = createWeight('n', 2, 'G0', 10, 'G1', 0.2, 'Gc', 1, 'w0', 6*2*pi) * ...
- createWeight('n', 2, 'G0', 1, 'G1', 5/0.2, 'Gc', 1/0.2, 'w0', 1300*2*pi);
-
-w_geo = createWeight('n', 2, 'G0', 0.6, 'G1', 0.2, 'Gc', 0.3, 'w0', 3*2*pi) * ...
- createWeight('n', 2, 'G0', 1, 'G1', 10/0.2, 'Gc', 1/0.2, 'w0', 800*2*pi);
-\end{minted}
-
-The measured dynamics are compared with the modelled one as well as the modelled uncertainty in Figure \ref{fig:dyn_uncertainty_acc} for the accelerometers and in Figure \ref{fig:dyn_uncertainty_geo} for the geophones.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/dyn_uncertainty_acc.png}
-\caption{\label{fig:dyn_uncertainty_acc}Modeled dynamical uncertainty and meaured dynamics of the accelerometers}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/dyn_uncertainty_geo.png}
-\caption{\label{fig:dyn_uncertainty_geo}Modeled dynamical uncertainty and meaured dynamics of the geophones}
-\end{figure}
-
-\subsection{\(\mathcal{H}_\infty\) Synthesis of Complementary Filters}
-\label{sec:org496909d}
-A last weight is now defined that represents the maximum dynamical uncertainty that is allowed for the super sensor.
-\begin{minted}[]{matlab}
-wu = inv(createWeight('n', 2, 'G0', 0.7, 'G1', 0.3, 'Gc', 0.4, 'w0', 3*2*pi) * ...
- createWeight('n', 2, 'G0', 1, 'G1', 6/0.3, 'Gc', 1/0.3, 'w0', 1200*2*pi));
-\end{minted}
-
-This dynamical uncertainty is compared with the two sensor uncertainties in Figure \ref{fig:uncertainty_weight_and_sensor_uncertainties}.
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/uncertainty_weight_and_sensor_uncertainties.png}
-\caption{\label{fig:uncertainty_weight_and_sensor_uncertainties}Individual sensor uncertainty (normalized by their dynamics) and the wanted maximum super sensor noise uncertainty}
-\end{figure}
-
-The generalized plant used for the synthesis is defined:
-\begin{minted}[]{matlab}
-P = [wu*w_acc -wu*w_acc;
- 0 wu*w_geo;
- 1 0];
-\end{minted}
-
-And the \(\mathcal{H}_\infty\) synthesis using the \texttt{hinfsyn} command is performed.
-\begin{minted}[]{matlab}
-[H_geo, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-\end{minted}
-
-\begin{verbatim}
-Test bounds: 0.8556 <= gamma <= 1.34
-
- gamma X>=0 Y>=0 rho(XY)<1 p/f
-1.071e+00 0.0e+00 0.0e+00 0.000e+00 p
-9.571e-01 0.0e+00 0.0e+00 9.436e-16 p
-9.049e-01 0.0e+00 0.0e+00 1.084e-15 p
-8.799e-01 0.0e+00 0.0e+00 1.191e-16 p
-8.677e-01 0.0e+00 0.0e+00 6.905e-15 p
-8.616e-01 0.0e+00 0.0e+00 0.000e+00 p
-8.586e-01 1.1e-17 0.0e+00 6.917e-16 p
-8.571e-01 0.0e+00 0.0e+00 6.991e-17 p
-8.564e-01 0.0e+00 0.0e+00 1.492e-16 p
-
-Best performance (actual): 0.8563
-\end{verbatim}
-
-The complementary filter is defined as follows:
-\begin{minted}[]{matlab}
-H_acc = 1 - H_geo;
-\end{minted}
-
-The bode plot of the obtained complementary filters is shown in Figure
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/h_infinity_obtained_complementary_filters.png}
-\caption{\label{fig:h_infinity_obtained_complementary_filters}Bode plot of the obtained complementary filters using the \(\mathcal{H}_\infty\) synthesis}
-\end{figure}
-
-\subsection{Obtained Super Sensor Dynamical Uncertainty}
-\label{sec:org8114f7c}
-The obtained super sensor dynamical uncertainty is shown in Figure \ref{fig:super_sensor_uncertainty_h_infinity}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/super_sensor_uncertainty_h_infinity.png}
-\caption{\label{fig:super_sensor_uncertainty_h_infinity}Obtained Super sensor dynamics uncertainty}
-\end{figure}
-
-\section{Optimal and Robust sensor fusion using the \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
-\label{sec:orgcd8e27b}
-\label{sec:optimal_sensor_fusion}
-\subsection{Noise and Dynamical uncertainty weights}
-\label{sec:org55c0ded}
-\begin{minted}[]{matlab}
-N_acc = (s/(2*pi*2000) + 1)^2/(s + 0.1*2*pi)/(s + 1e3*2*pi)/(1 + s/2/pi/1e3); % [m/sqrt(Hz)]
-N_geo = 4e-4*((s + 2*pi)/(2*pi*200) + 1)/(s + 1e3*2*pi)/(1 + s/2/pi/1e3); % [m/sqrt(Hz)]
-\end{minted}
-
-\begin{minted}[]{matlab}
-w_acc = createWeight('n', 2, 'G0', 10, 'G1', 0.2, 'Gc', 1, 'w0', 6*2*pi) * ...
- createWeight('n', 2, 'G0', 1, 'G1', 5/0.2, 'Gc', 1/0.2, 'w0', 1300*2*pi);
-
-w_geo = createWeight('n', 2, 'G0', 0.6, 'G1', 0.2, 'Gc', 0.3, 'w0', 3*2*pi) * ...
- createWeight('n', 2, 'G0', 1, 'G1', 10/0.2, 'Gc', 1/0.2, 'w0', 800*2*pi);
-\end{minted}
-
-\begin{minted}[]{matlab}
-wu = inv(createWeight('n', 2, 'G0', 0.7, 'G1', 0.3, 'Gc', 0.4, 'w0', 3*2*pi) * ...
- createWeight('n', 2, 'G0', 1, 'G1', 6/0.3, 'Gc', 1/0.3, 'w0', 1200*2*pi));
-\end{minted}
-
-\begin{minted}[]{matlab}
-P = [wu*w_acc -wu*w_acc;
- 0 wu*w_geo;
- N_acc -N_acc;
- 0 N_geo;
- 1 0];
-\end{minted}
-
-And the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed.
-\begin{minted}[]{matlab}
-[H_geo, ~] = h2hinfsyn(ss(P), 1, 1, 2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 1e-3, 'DISPLAY', 'on');
-\end{minted}
-
-\begin{minted}[]{matlab}
-H_acc = 1 - H_geo;
-\end{minted}
-
-\subsection{Obtained Super Sensor Noise}
-\label{sec:org925fb2f}
-\begin{minted}[]{matlab}
-freqs = logspace(0, 4, 1000);
-PSD_Sgeo = abs(squeeze(freqresp(N_geo, freqs, 'Hz'))).^2;
-PSD_Sacc = abs(squeeze(freqresp(N_acc, freqs, 'Hz'))).^2;
-PSD_Hss = abs(squeeze(freqresp(N_acc*H_acc, freqs, 'Hz'))).^2 + ...
- abs(squeeze(freqresp(N_geo*H_geo, freqs, 'Hz'))).^2;
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/psd_sensors_htwo_hinf_synthesis.png}
-\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}
-
-\subsection{Obtained Super Sensor Dynamical Uncertainty}
-\label{sec:org3ca6152}
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.png}
-\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}
-
-\subsection{Experimental Super Sensor Dynamical Uncertainty}
-\label{sec:org9ea8a83}
-The super sensor dynamics is shown in Figure \ref{fig:super_sensor_optimal_uncertainty}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/super_sensor_optimal_uncertainty.png}
-\caption{\label{fig:super_sensor_optimal_uncertainty}Inertial Sensor dynamics as well as the super sensor dynamics}
-\end{figure}
-
-\subsection{Experimental Super Sensor Noise}
-\label{sec:orgfacb28c}
-The obtained super sensor noise is shown in Figure \ref{fig:super_sensor_optimal_noise}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/super_sensor_optimal_noise.png}
-\caption{\label{fig:super_sensor_optimal_noise}ASD of the super sensor obtained using the \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
-\end{figure}
-
-\section{Matlab Functions}
-\label{sec:org95268dc}
-\label{sec:matlab_functions}
-\subsection{\texttt{createWeight}}
-\label{sec:org5453f8d}
-\label{sec:createWeight}
-
-This Matlab function is accessible \href{src/createWeight.m}{here}.
-
-\begin{minted}[]{matlab}
-function [W] = createWeight(args)
-% createWeight -
-%
-% Syntax: [in_data] = createWeight(in_data)
-%
-% Inputs:
-% - n - Weight Order
-% - G0 - Low frequency Gain
-% - G1 - High frequency Gain
-% - Gc - Gain of W at frequency w0
-% - w0 - Frequency at which |W(j w0)| = Gc
-%
-% Outputs:
-% - W - Generated Weight
-
- arguments
- args.n (1,1) double {mustBeInteger, mustBePositive} = 1
- args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
- args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
- args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
- args.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
- end
-
- mustBeBetween(args.G0, args.Gc, args.G1);
-
- s = tf('s');
-
- W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
-
- end
-
- % Custom validation function
- function mustBeBetween(a,b,c)
- if ~((a > b && b > c) || (c > b && b > a))
- eid = 'createWeight:inputError';
- msg = 'Gc should be between G0 and G1.';
- throwAsCaller(MException(eid,msg))
- end
- end
-\end{minted}
-
-\subsection{\texttt{plotMagUncertainty}}
-\label{sec:orgc63fba3}
-\label{sec:plotMagUncertainty}
-
-This Matlab function is accessible \href{src/plotMagUncertainty.m}{here}.
-
-\begin{minted}[]{matlab}
-function [p] = plotMagUncertainty(W, freqs, args)
-% plotMagUncertainty -
-%
-% Syntax: [p] = plotMagUncertainty(W, freqs, args)
-%
-% Inputs:
-% - W - Multiplicative Uncertainty Weight
-% - freqs - Frequency Vector [Hz]
-% - args - Optional Arguments:
-% - G
-% - color_i
-% - opacity
-%
-% Outputs:
-% - p - Plot Handle
-
-arguments
- W
- freqs double {mustBeNumeric, mustBeNonnegative}
- args.G = tf(1)
- args.color_i (1,1) double {mustBeInteger, mustBePositive} = 1
- args.opacity (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.3
- args.DisplayName char = ''
-end
-
-% Get defaults colors
-colors = get(groot, 'defaultAxesColorOrder');
-
-p = patch([freqs flip(freqs)], ...
- [abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*(1 + abs(squeeze(freqresp(W, freqs, 'Hz')))); ...
- flip(abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*max(1 - abs(squeeze(freqresp(W, freqs, 'Hz'))), 1e-6))], 'w', ...
- 'DisplayName', args.DisplayName);
-
-p.FaceColor = colors(args.color_i, :);
-p.EdgeColor = 'none';
-p.FaceAlpha = args.opacity;
-
-end
-\end{minted}
-
-\subsection{\texttt{plotPhaseUncertainty}}
-\label{sec:orga4018d2}
-\label{sec:plotPhaseUncertainty}
-
-This Matlab function is accessible \href{src/plotPhaseUncertainty.m}{here}.
-
-\begin{minted}[]{matlab}
-function [p] = plotPhaseUncertainty(W, freqs, args)
-% plotPhaseUncertainty -
-%
-% Syntax: [p] = plotPhaseUncertainty(W, freqs, args)
-%
-% Inputs:
-% - W - Multiplicative Uncertainty Weight
-% - freqs - Frequency Vector [Hz]
-% - args - Optional Arguments:
-% - G
-% - color_i
-% - opacity
-%
-% Outputs:
-% - p - Plot Handle
-
-arguments
- W
- freqs double {mustBeNumeric, mustBeNonnegative}
- args.G = tf(1)
- args.color_i (1,1) double {mustBeInteger, mustBePositive} = 1
- args.opacity (1,1) double {mustBeNumeric, mustBePositive} = 0.3
- args.DisplayName char = ''
-end
-
-% Get defaults colors
-colors = get(groot, 'defaultAxesColorOrder');
-
-% Compute Phase Uncertainty
-Dphi = 180/pi*asin(abs(squeeze(freqresp(W, freqs, 'Hz'))));
-Dphi(abs(squeeze(freqresp(W, freqs, 'Hz'))) > 1) = 360;
-
-% Compute Plant Phase
-G_ang = 180/pi*angle(squeeze(freqresp(args.G, freqs, 'Hz')));
-
-p = patch([freqs flip(freqs)], [G_ang+Dphi; flip(G_ang-Dphi)], 'w', ...
- 'DisplayName', args.DisplayName);
-
-p.FaceColor = colors(args.color_i, :);
-p.EdgeColor = 'none';
-p.FaceAlpha = args.opacity;
-
-end
-\end{minted}
-
-\bibliography{ref}
-
-\printbibliography
-\end{document}
diff --git a/js/bootstrap.min.js b/js/bootstrap.min.js
deleted file mode 100644
index c8f82e5..0000000
--- a/js/bootstrap.min.js
+++ /dev/null
@@ -1,7 +0,0 @@
-/*!
- * Bootstrap v3.3.4 (http://getbootstrap.com)
- * Copyright 2011-2015 Twitter, Inc.
- * Licensed under MIT (https://github.com/twbs/bootstrap/blob/master/LICENSE)
- */
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