Commit d0086204 authored by Pierre-antoine Comby's avatar Pierre-antoine Comby

initial commit

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## Core latex/pdflatex auxiliary files:
*.aux
*.lof
*.log
*.lot
*.fls
*.out
*.toc
*.fmt
*.fot
*.cb
*.cb2
.*.lb
## Intermediate documents:
*.dvi
*.xdv
*-converted-to.*
# these rules might exclude image files for figures etc.
# *.ps
# *.eps
*.pdf
## Generated if empty string is given at "Please type another file name for output:"
.pdf
## Bibliography auxiliary files (bibtex/biblatex/biber):
*.bbl
*.bcf
*.blg
*-blx.aux
*-blx.bib
*.run.xml
## Build tool auxiliary files:
*.fdb_latexmk
*.synctex
*.synctex(busy)
*.synctex.gz
*.synctex.gz(busy)
*.pdfsync
## Build tool directories for auxiliary files
# latexrun
latex.out/
## Auxiliary and intermediate files from other packages:
# algorithms
*.alg
*.loa
# achemso
acs-*.bib
# amsthm
*.thm
# beamer
*.nav
*.pre
*.snm
*.vrb
# changes
*.soc
# comment
*.cut
# cprotect
*.cpt
# elsarticle (documentclass of Elsevier journals)
*.spl
# endnotes
*.ent
# fixme
*.lox
# feynmf/feynmp
*.mf
*.mp
*.t[1-9]
*.t[1-9][0-9]
*.tfm
#(r)(e)ledmac/(r)(e)ledpar
*.end
*.?end
*.[1-9]
*.[1-9][0-9]
*.[1-9][0-9][0-9]
*.[1-9]R
*.[1-9][0-9]R
*.[1-9][0-9][0-9]R
*.eledsec[1-9]
*.eledsec[1-9]R
*.eledsec[1-9][0-9]
*.eledsec[1-9][0-9]R
*.eledsec[1-9][0-9][0-9]
*.eledsec[1-9][0-9][0-9]R
# glossaries
*.acn
*.acr
*.glg
*.glo
*.gls
*.glsdefs
# gnuplottex
*-gnuplottex-*
# gregoriotex
*.gaux
*.gtex
# htlatex
*.4ct
*.4tc
*.idv
*.lg
*.trc
*.xref
# hyperref
*.brf
# knitr
*-concordance.tex
# TODO Comment the next line if you want to keep your tikz graphics files
*.tikz
*-tikzDictionary
# listings
*.lol
# makeidx
*.idx
*.ilg
*.ind
*.ist
# minitoc
*.maf
*.mlf
*.mlt
*.mtc[0-9]*
*.slf[0-9]*
*.slt[0-9]*
*.stc[0-9]*
# minted
_minted*
*.pyg
# morewrites
*.mw
# nomencl
*.nlg
*.nlo
*.nls
# pax
*.pax
# pdfpcnotes
*.pdfpc
# sagetex
*.sagetex.sage
*.sagetex.py
*.sagetex.scmd
# scrwfile
*.wrt
# sympy
*.sout
*.sympy
sympy-plots-for-*.tex/
# pdfcomment
*.upa
*.upb
# pythontex
*.pytxcode
pythontex-files-*/
# tcolorbox
*.listing
# thmtools
*.loe
# TikZ & PGF
*.dpth
*.md5
*.auxlock
# todonotes
*.tdo
# vhistory
*.hst
*.ver
# easy-todo
*.lod
# xcolor
*.xcp
# xmpincl
*.xmpi
# xindy
*.xdy
# xypic precompiled matrices
*.xyc
# endfloat
*.ttt
*.fff
# Latexian
TSWLatexianTemp*
## Editors:
# WinEdt
*.bak
*.sav
# Texpad
.texpadtmp
# LyX
*.lyx~
# Kile
*.backup
# KBibTeX
*~[0-9]*
# auto folder when using emacs and auctex
./auto/*
*.el
# expex forward references with \gathertags
*-tags.tex
# standalone packages
*.sta
\documentclass[12pt,a4paper,french]{article}
\usepackage[utf8x]{inputenc}
\usepackage{ucs}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{makeidx}
\usepackage{graphicx}
\usepackage{babel}
\usepackage[left=2.00cm, right=2.00cm, top=3.00cm, bottom=3.00cm]{geometry}
\usepackage{fancyhdr}
\usepackage{float}
\usepackage{subfig}
\usepackage[dvipsnames,x11names]{xcolor}
\usepackage{listingsutf8}
\usepackage[section]{placeins}
\author{Pierre-Antoine \textsc{Comby}, Thomas \textsc{Bauvent}}
\title{TP - Identification de systèmes et annulation d'écho}
\date{}
\renewcommand{\thesection}{\Alph{section}}
\renewcommand{\thesubsection}{\Roman{subsection}}
\newcommand{\deriv}[2][]{\frac{\d#1}{\d#2}}
\newcommand{\derivp}[2][]{\frac{\partial#1}{\partial#2}}
\newcommand{\derivpp}[2][]{\frac{\partial^2#1}{\partial#2^2}}
\newenvironment{syslin}[1][l]{\ensuremath\left\lbrace\begin{array}{#1}}{\end{array}\right.}% © PAC ;)
\let\vec\mathbf
\begin{document}
\maketitle
\section{Préparation}
\begin{enumerate}
\item
\begin{enumerate}
\item La formule générale est $\vec{s}= \vec{H} \vec{e} + \vec{b}$
Et pour une composante du vecteur s on a : \[ s_n = \sum_{m=0}^{L1} h_m e_{n-m} + b_n\]
\item
C'est une convolution discrète de $\vec{h}$ et $\vec{e}$ qui traduit l'influence de tout le signal déja transmis sur le signal reçu à l'instant $n$.
\item $L$ est la longueur du canal , $ie$ le temps d'observation du signal
\end{enumerate}
\item Le modèle est ici affine (LP et LE)
\item
\[
\begin{pmatrix}
s_{L-1}\\
s_L\\
\vdots\\
s_{M-1}
\end{pmatrix}
= \underbrace{\begin{pmatrix}
e_{L-1} & e_{L-2} & \dots & e_0 \\
e_L & e_{L-1} & \dots & e_1 \\
\vdots & \vdots & \dots & \vdots \\
e_{M-1} & e_{M-2} & \dots & e_{M-L}
\end{pmatrix}}_{\vec{E}\in \mathcal{M}_{M-L+1,L}}
\begin{pmatrix}
h_{L-1}\\
h_L\\
\vdots\\
h_{M-1}
\end{pmatrix} +
\begin{pmatrix}
b_{L-1}\\
b_L\\
\vdots\\
b_{M-1}
\end{pmatrix}
\]
\item En connaissant les $e_n$ pour $n<0$ on considère alors un canal moyen, qui n'est plus image du comportement actuel.
\item \[\hat{\vec{h}}_{MC} = \arg\min_{\vec{h}}\underbrace{\|\vec{s_a}-\vec{E}\vec{h}\|^2}_{\phi}\]
On veux donc :
\begin{align*}
\derivp[\phi]{\vec{h}} &= 0 \\
-2\vec{E}^T\vec{s_a} + 2 \vec{E}^T\vec{E}\vec{h} &= 0 \\
\hat{\vec{h}_{MC}} &= \left(\vec{E}^T\vec{E}\right)^{-1}\vec{E}^T\vec{s_a}
\end{align*}
\item on a la matrice $\vec{E}^T\vec{E}$ de taille $L\times L $Pour toruver $h_{MC}$ on résoudra plutot le système linéaire , moins couteux en calcul que d'inverser une matrice.
\item De même que précedement (la convolution est symétrique)
\[
\begin{pmatrix}
s_{L-1}\\
s_L\\
\vdots\\
s_{M-1}
\end{pmatrix}
= \underbrace{\begin{pmatrix}
h_{L-1} & e_{L-2} & \dots & h_{M+L-N} \\
h_L & h_{L-1} & \dots & h_{L+M-N+1} \\
\vdots & \vdots & \dots & \vdots \\
h_{N-1-M} & \dots & \dots & h_{0}
\end{pmatrix}}_{\vec{E}\in \mathcal{M}_{M-L+1,L}}
\begin{pmatrix}
e_{M}\\
\vdots\\
\vdots\\
e_{N-1}
\end{pmatrix} +
\begin{pmatrix}
b_{M}\\
\vdots\\
\vdots\\
b_{N-1}
\end{pmatrix}
\]
\item de même qu'en 5. et on obtient :
\item
\[ \hat{\vec{e}_{MC}} = \left(\vec{H}^T\vec{H}\right)^{-1}\vec{H}^T\vec{s_a}\]
\end{enumerate}
\section{Programmation de l'optimisation}
\emph{Le code source du TP est joint en annexe}
\paragraph{12.} ~
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\linewidth]{signal}
\caption{Signal reçu et déconvolués}
\label{fig:signal}
\end{figure}
\paragraph{13.} On s'attend à obtenir trois diracs , images des différents echos du système.
\paragraph{15.} On a une structure de \textsc{Toeplitz}.
\paragraph{17.} ~
\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{h_N_500_rcond_0_0012}
\caption{$\vec{h_{MC}}$ en utilisant la totalité du signal.}
\label{fig:hn500rcond00012}
\end{figure}
\paragraph{18.}~
\begin{figure}
\centering
\subfloat[$N=450, cond=0.0014$]{\includegraphics[width=0.3\linewidth]{h_N_450_rcond_0_0014.png}}
\subfloat[$N=350, cond=0.0016$]{\includegraphics[width=0.3\linewidth]{h_N_350_rcond_0_0016.png}}\\
\subfloat[$N=200, cond=0.0017$]{\includegraphics[width=0.3\linewidth]{h_N_200_rcond_0_0027.png}}
\subfloat[$N=500, cond=0.0012$]{\includegraphics[width=0.3\linewidth]{h_N_500_rcond_0_0012.png}}
\caption{Valeurs de $\vec{h}$ en fonction de la quantité de signal utilisé et conditionnement de la matrice associée.}
\end{figure}
\end{document}
\documentclass[12pt,a4paper,french]{article}
\usepackage[utf8x]{inputenc}
\usepackage{mathtools}
\usepackage{makeidx}
\usepackage{graphicx}
\usepackage{babel}
\usepackage[left=2.00cm, right=2.00cm, top=3.00cm, bottom=3.00cm]{geometry}
\usepackage{float}
\usepackage{caption}
\usepackage{subcaption}
\usepackage[dvipsnames,x11names]{xcolor}
\usepackage{listingsutf8}
\usepackage[section]{placeins}
\author{Pierre-Antoine \textsc{Comby}, Thomas \textsc{Bauvent}}
\title{TP - Estimation de la viscosité d'un fluide}
\date{}
\newcommand{\deriv}[2][]{\ensuremath\frac{d#1}{d#2}}
\newcommand{\derivp}[2][]{\ensuremath\frac{\partial#1}{\partial#2}}
\newcommand{\derivpp}[2][]{\ensuremath\frac{\partial^2#1}{\partial#2^2}}
\newenvironment{syslin}[1][l]{\ensuremath\left\lbrace\begin{array}{#1}}{\end{array}\right.}% © PAC ;)
\let\vec\mathbf
\begin{document}
\maketitle
\section{Préparation}
\begin{enumerate}
\item On a à partir du PFD sur la bille :
\[m\ddot{h} = (\rho_b-\rho_l)\frac{4}{3}\pi R^3_b g + k|\dot{h}|^n\]
Que l'on met en forme ( $m = \rho_b \frac{4}{3}\pi R^3_b$) :
\[\ddot{h}+\frac{k}{m}|\dot{h}| =\left(1-\frac{\rho_l}{\rho_b}\right)g\]
Avec :
$\begin{dcases}
h(0) = H \\
\dot{h}(0) =0
\end{dcases}$
\item On introduit donc le vecteur d'état :$\vec{x} = \left(
\begin{array}{l}
h \\
\dot{h}
\end{array}\right)
=\left(\begin{array}{l}
x_1\\
x_2
\end{array}\right)$
et on pose également $\vec{x}(0) = \vec{x_0}= \left(\begin{array}{l}
H\\
0
\end{array}\right)$
On a donc :
\[
\deriv[\vec{x}]{t} = \left(\begin{array}{c}
x_2\\
\displaystyle
\left(\frac{\rho_l}{\rho_b}-1\right)g + \frac{k}{m}|x_2|^n
\end{array}\right)
\]
\item On utilise dans la suite le critère des moindres carrés:
\[
c(p) = \|\vec{h_m}-\vec{h}\|^2_2= \sum_{i=0}^{N-1}\left(h_ {mod}(t_i)-h_{mes}(t_i)\right)^2
\]
\item
\[
\vec{g}(\vec{p})= \left(\begin{array}{c}
\displaystyle
\derivp[c(\vec{p})]{p_1}\\
\displaystyle
\derivp[c(\vec{p})]{p_2}\\
\end{array}\right)
\]
Avec
\[
\begin{dcases}
\displaystyle
\derivp[c(\vec{p})]{p_1} =
\sum_{i=0}^{N-1} 2 \derivp[h_{mod}]{p_1}(t_i)
\left(h_{mod}(t_i)-h_{mes}(t_i)\right)
= \sum_{i