From 341b6531011ad16242ab766e8c0375afe52527b4 Mon Sep 17 00:00:00 2001 From: Tournesol02 Date: Thu, 7 May 2026 19:58:41 +0200 Subject: [PATCH] Changement de la matrice B --- latex/Questions/Q13.tex | 2 +- latex/Questions/Q17.log | 140 +++++++++++++++++++++++++++++++--------- latex/Questions/Q17.tex | 30 +++++---- latex/Questions/Q3.tex | 18 +++--- latex/Questions/Q4.tex | 31 ++++----- latex/main.aux | 5 +- latex/main.log | 130 +++++++++++++++++++------------------ latex/main.pdf | Bin 658411 -> 657816 bytes latex/main.synctex.gz | Bin 52650 -> 50234 bytes 9 files changed, 217 insertions(+), 139 deletions(-) diff --git a/latex/Questions/Q13.tex b/latex/Questions/Q13.tex index 3379f1f..6f325a2 100644 --- a/latex/Questions/Q13.tex +++ b/latex/Questions/Q13.tex @@ -4,7 +4,7 @@ Avec le même periode d'echantillonnage comme dans la section precedente: $T_e=0 \begin{figure}[H] \centering - \includegraphics[scale=0.3]{Illustrations/NouveauxImages/Q13} + \includegraphics[scale=0.5]{Illustrations/NouveauxImages/Q13} \caption{Pôles du système avec temps d'echantillonnage: $T_e = 0.0356$} \label{fig:q13} \end{figure} diff --git a/latex/Questions/Q17.log b/latex/Questions/Q17.log index 80b4dac..c151b70 100644 --- a/latex/Questions/Q17.log +++ b/latex/Questions/Q17.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.4.14) 7 MAY 2026 19:29 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.4.14) 7 MAY 2026 19:39 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -339,30 +339,9 @@ Overfull \hbox (4.44444pt too wide) in paragraph at lines 3--5 \/cmr/m/n/10 ^^Se [] - -! LaTeX Error: Missing \begin{document}. - -See the LaTeX manual or LaTeX Companion for explanation. -Type H for immediate help. - ... - -l.21 p - _{\text{des}} = -You're in trouble here. Try typing to proceed. -If that doesn't work, type X to quit. - -Missing character: There is no p in font nullfont! -! Missing $ inserted. - - $ -l.21 p_ - {\text{des}} = -I've inserted a begin-math/end-math symbol since I think -you left one out. Proceed, with fingers crossed. - ! Undefined control sequence. -l.21 p_{\text - {des}} = +l.6 $p_{\text + {des}} =$ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct @@ -370,31 +349,128 @@ spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. +Overfull \hbox (27.81807pt too wide) in paragraph at lines 6--7 + \OML/cmm/m/it/10 p[] \/cmr/m/n/10 =$ + [] + + ! LaTeX Error: Environment bmatrix undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... -l.22 \begin{bmatrix} - +l.8 \begin{bmatrix} + Your command was ignored. Type I to replace it with another command, or to continue without it. +! Misplaced alignment tab character &. +l.9 -85 & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. -! LaTeX Error: \begin{document} ended by \end{bmatrix}. +! Misplaced alignment tab character &. +l.10 -8+69i & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.11 -8-69i & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.12 -28+17i & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.13 -28-17i & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.14 -16 & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.15 -4+3i & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.16 -4-3i & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 7 ended by \end{bmatrix}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... -l.32 \end{bmatrix} +l.18 \end{bmatrix} ^{T} Your command was ignored. 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Here is how much of TeX's memory you used: - 37 strings out of 474222 - 630 string characters out of 5748733 + 39 strings out of 474222 + 690 string characters out of 5748733 1923975 words of memory out of 5000000 - 22390 multiletter control sequences out of 15000+600000 + 22392 multiletter control sequences out of 15000+600000 558377 words of font info for 37 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 - 23i,3n,32p,268b,91s stack positions out of 10000i,1000n,20000p,200000b,200000s + 23i,3n,32p,268b,133s stack positions out of 10000i,1000n,20000p,200000b,200000s ! ==> Fatal error occurred, no output PDF file produced! diff --git a/latex/Questions/Q17.tex b/latex/Questions/Q17.tex index 8674fbf..1cab950 100644 --- a/latex/Questions/Q17.tex +++ b/latex/Questions/Q17.tex @@ -18,18 +18,20 @@ $p_{\text{des}} =$ \end{bmatrix}^{T} \end{equation} -\begin{equation} -p_{\text{des}} = -\begin{bmatrix} --84.6647 \\ --7.5585+68.7669i \\ --7.5585-68.7669i \\ --27.9691+17.2408i \\ --27.9691-17.2408i \\ --15.9108 \\ --4.3805+2.6626i \\ --4.3805-2.6626i \\ --4.3805 -\end{bmatrix} -\end{equation} +%\begin{equation} +%p_{\text{des}} = +%\begin{bmatrix} +%-84.6647 \\ +%-7.5585+68.7669i \\ +%-7.5585-68.7669i \\ +%-27.9691+17.2408i \\ +%-27.9691-17.2408i \\ +%-15.9108 \\ +%-4.3805+2.6626i \\ +%-4.3805-2.6626i \\ +%-4.3805 +%\end{bmatrix} +%\end{equation} + + \end{center} \ No newline at end of file diff --git a/latex/Questions/Q3.tex b/latex/Questions/Q3.tex index 8a29caa..7c4897d 100644 --- a/latex/Questions/Q3.tex +++ b/latex/Questions/Q3.tex @@ -46,15 +46,17 @@ B= \end{bmatrix} = \begin{bmatrix} -0\\ -0\\ -0\\ -0\\ -4\\ --16\\ --60\\ +0 & +0 & +0 & +0 & +4 & +-16 & +-60 & -120 -\end{bmatrix} +\end{bmatrix}^{T} +\end{equation} +\begin{equation} \qquad C= \begin{bmatrix} diff --git a/latex/Questions/Q4.tex b/latex/Questions/Q4.tex index 95ed348..b927508 100644 --- a/latex/Questions/Q4.tex +++ b/latex/Questions/Q4.tex @@ -1,40 +1,35 @@ \textbf{Question 4} -La Figure 1a correspond aux valeurs propres en temps continu, tandis que les figures \textit{1b, 1c et 1d} montrent les valeurs propres discrètes obtenues avec les méthodes de Tustin et ZOH pour différents temps d'échantillonnage. - -\begin{figure}[h] - \centering +Voici le lieu des racines du système en continu et en temps discret. $T_s = {0.01 ; 0.02 ; 0.04}$ avec le cercle unitaire. Comme tous nos pôles sont à $|z|=1$, nous ne pouvons pas conclure sur la stabilité ni l'instabilité. En temps continu nous avons instabilité, à cause des pôles: $Re\{\lambda\}>0$ +\begin{figure}[H] +\centering \begin{subfigure}{0.48\textwidth} \centering - \includegraphics[width=\textwidth]{Illustrations/Q4eigenValTEMPSCONT} + \includegraphics[scale=0.3]{Illustrations/Q4eigenValTEMPSCONT} \caption{Valeurs propres en temps continu.} \end{subfigure} \hfill \begin{subfigure}{0.48\textwidth} \centering - \includegraphics[width=\textwidth]{Illustrations/Q40100} - \caption{$T_s = 0{,}01\,\mathrm{s}$.} + \includegraphics[scale=0.3]{Illustrations/Q40100} + \caption{$T_s = 0.01\,\mathrm{s}$.} \end{subfigure} - - \vspace{0.3cm} - - \begin{subfigure}{0.48\textwidth} + \begin{subfigure}{0.48\textwidth} \centering - \includegraphics[width=\textwidth]{Illustrations/Q40200} - \caption{$T_s = 0{,}02\,\mathrm{s}$.} + \includegraphics[scale=0.3]{Illustrations/Q40200} + \caption{$T_s = 0.02\,\mathrm{s}$.} \end{subfigure} \hfill \begin{subfigure}{0.48\textwidth} \centering - \includegraphics[width=\textwidth]{Illustrations/Q40400} - \caption{$T_s = 0{,}04\,\mathrm{s}$.} + \includegraphics[scale=0.3]{Illustrations/Q40400} + \caption{$T_s = 0.04\,\mathrm{s}$.} \end{subfigure} - \caption{Comparaison des valeurs propres du système en temps continu et en temps discret.} + \label{fig:q9_eigenvalues} \end{figure} -En temps continu, le système présente deux pôles à partie réelle positive, ce qui montre son instabilité. -Après discrétisation, les pôles obtenus avec les méthodes de Tustin et ZOH se trouvent sur ou proches du cercle unité. Comme tous nos pôles sont à $|z|=1$, nous ne pouvons pas conclure sur la stabilité ni l'instabilité. + diff --git a/latex/main.aux b/latex/main.aux index dcadd29..1b1507b 100644 --- a/latex/main.aux +++ b/latex/main.aux @@ -15,10 +15,11 @@ \@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Dynamique avec EDP}{1}{subsection.1.1}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Discrétisation par éléments finis}{1}{subsection.1.2}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Comparaison des valeurs propres du système en temps continu et en temps discret.}}{3}{figure.caption.1}\protected@file@percent } -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Response du système en boucle ouverte.}}{3}{figure.caption.2}\protected@file@percent } +\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} +\newlabel{fig:q9_eigenvalues}{{1}{3}{Comparaison des valeurs propres du système en temps continu et en temps discret}{figure.caption.1}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Response du système en boucle ouverte.}}{4}{figure.caption.2}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {2}Retour de sortie}{4}{section.2}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Suivi de la consigne $w_c(t)$ par la position $w(L,t)$.}}{4}{figure.caption.3}\protected@file@percent } -\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} \newlabel{fig:Q8_1}{{3}{4}{Suivi de la consigne $w_c(t)$ par la position $w(L,t)$}{figure.caption.3}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Retour de sortie par ZOH}{4}{subsection.2.1}\protected@file@percent } \newlabel{fig:q9_eigenvalues}{{\caption@xref {fig:q9_eigenvalues}{ on input line 30}}{5}{Retour de sortie par ZOH}{figure.caption.4}{}} diff --git a/latex/main.log b/latex/main.log index 5250fb3..b9ee2ab 100644 --- a/latex/main.log +++ b/latex/main.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.4.14) 7 MAY 2026 19:33 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.4.14) 7 MAY 2026 19:57 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -1067,7 +1067,11 @@ LaTeX Font Info: Trying to load font information for T1+phv on input line 50 (/usr/share/texlive/texmf-dist/tex/latex/psnfss/t1phv.fd File: t1phv.fd 2020/03/25 scalable font definitions for T1/phv. -) (./main.aux) +) (./main.aux + +LaTeX Warning: Label `fig:q9_eigenvalues' multiply defined. + +) \openout1 = `main.aux'. 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