From eb8f8d1cdddad2e5c11dd574f07424e5da7dd156 Mon Sep 17 00:00:00 2001 From: Tournesol02 Date: Thu, 7 May 2026 19:35:21 +0200 Subject: [PATCH] =?UTF-8?q?Dernieres=20images=20ajout=C3=A9s?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- latex/Questions/Q10.tex | 19 +- latex/Questions/Q11.tex | 2 +- latex/Questions/Q13.tex | 10 +- latex/Questions/Q14.tex | 29 ++- latex/Questions/Q15.tex | 4 +- latex/Questions/Q16.tex | 2 +- latex/Questions/Q17.log | 413 ++++++++++++++++++++++++++++++++++++++++ latex/Questions/Q17.tex | 37 ++-- latex/Questions/Q18.tex | 4 +- latex/Questions/Q5.log | 335 ++++++++++++++++++++++++++++++++ latex/Questions/Q5.tex | 2 +- latex/Questions/Q7.tex | 2 +- latex/Questions/Q8.tex | 2 +- latex/Questions/Q9.tex | 29 ++- latex/contents.tex | 4 - latex/main.aux | 41 ++-- latex/main.log | 266 +++++++++++++++----------- latex/main.pdf | Bin 560281 -> 658411 bytes latex/main.stc2 | 2 +- latex/main.stc3 | 2 +- latex/main.stc4 | 2 +- latex/main.synctex.gz | Bin 51154 -> 52650 bytes latex/main.tex | 2 +- latex/main.toc | 10 +- 24 files changed, 1018 insertions(+), 201 deletions(-) create mode 100644 latex/Questions/Q17.log create mode 100644 latex/Questions/Q5.log diff --git a/latex/Questions/Q10.tex b/latex/Questions/Q10.tex index 09933ab..7179680 100644 --- a/latex/Questions/Q10.tex +++ b/latex/Questions/Q10.tex @@ -1,12 +1,21 @@ \textbf{Question 10} -Nous avons trouvé un seuil de $T_s = 0.0356s$ à l'aide d'un boucle qui traverse les +Nous avons trouvé un seuil de $T_s = 0.0356s$ à l'aide d'un boucle qui traverse les differents periodes d'echantillonnage. + \begin{figure}[H] - \centering - \includegraphics[scale=0.4]{Illustrations/Q10} - \caption{Système à $T_s = 2s$} - \label{fig:Q10_1} +\centering + \begin{subfigure}{0.48\textwidth} + \centering + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q10} + \caption{Système au seuil: $T_s = 0.0356s$} + \end{subfigure} + \hfill + \begin{subfigure}{0.48\textwidth} + \centering + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q10_diverge} + \caption{Système hors seuil: $T_s = 0.0366s$} + \end{subfigure} \end{figure} diff --git a/latex/Questions/Q11.tex b/latex/Questions/Q11.tex index 28bca7e..563c0ac 100644 --- a/latex/Questions/Q11.tex +++ b/latex/Questions/Q11.tex @@ -5,6 +5,6 @@ Avec la fonction \texttt{lqr} de MATLAB, le gain de retour d'état obtenu est : \begin{equation} K = \begin{bmatrix} -0{,}97 & -14{,}62 & 0{,}66 & 1{,}32 & 19{,}32 & 0{,}39 & 2{,}40 & -2{,}11 +0.97 & -14.62 & 0{,}66 & 1.32 & 19.32 & 0.39 & 2.40 & -2.11 \end{bmatrix} \end{equation} \ No newline at end of file diff --git a/latex/Questions/Q13.tex b/latex/Questions/Q13.tex index f940fd9..3379f1f 100644 --- a/latex/Questions/Q13.tex +++ b/latex/Questions/Q13.tex @@ -1,12 +1,14 @@ \textbf{Question 13} -Nous avons testé plusieurs périodes d'échantillonnage entre $0{,}1\,\mathrm{s}$ et $5\,\mathrm{s}$. -Les valeurs propres discrètes restent à l'intérieur du cercle unité, comme le montre la Figure~\ref{fig:q13}. +Avec le même periode d'echantillonnage comme dans la section precedente: $T_e=0.0356$ notre système est rendu instable. \begin{figure}[H] \centering - \includegraphics[width=0.4\textwidth]{Illustrations/im1} % Remplacez par votre image - \caption{Valeurs propres pour différentes périodes d'échantillonnage.} + \includegraphics[scale=0.3]{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/Q14.tex b/latex/Questions/Q14.tex index ca82350..0e2a202 100644 --- a/latex/Questions/Q14.tex +++ b/latex/Questions/Q14.tex @@ -1,10 +1,29 @@ \textbf{Question 14} -Comme nous avions fait en Q12. Par approche empirique, nous avons trouvé que $T_s = 0.3s$, ce qui convient pour maintenir le comportement temporel voulu. Voir Figure~\ref{fig:q14_reponses}. +Nous retrouvons un seuil d'echantillonnage à: $T_e = 0.0107s$. Voici les comportements du système dedans et hors seuil. \begin{figure}[H] - \centering - \includegraphics[width=0.4\textwidth]{Illustrations/Q14} % Remplacez par votre image - \caption{Réponse du système à $T_s = 0.3s$.} - \label{fig:q14_reponses} +\centering + \begin{subfigure}{0.48\textwidth} + \centering + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q14} + \caption{Système au seuil: $T_s = 0.0107s$} + \end{subfigure} + \hfill + \begin{subfigure}{0.48\textwidth} + \centering + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q14_instable} + \caption{Système hors seuil: $T_s = 0.0108s$} + \end{subfigure} + \begin{subfigure}{0.48\textwidth} + \centering + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q14_step} + \caption{Reponse à un echelon unitaire: $T_s = 0.0107s$} + \end{subfigure} + \hfill + \begin{subfigure}{0.48\textwidth} + \centering + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q14_step_diverge} + \caption{Reponse à un echelon unitaire: $T_s = 0.0108s$} + \end{subfigure} \end{figure} \ No newline at end of file diff --git a/latex/Questions/Q15.tex b/latex/Questions/Q15.tex index ccb0a3e..da62997 100644 --- a/latex/Questions/Q15.tex +++ b/latex/Questions/Q15.tex @@ -7,7 +7,7 @@ B_p = \end{bmatrix} = \begin{bmatrix} -0 \\ 0 \\ 0 \\ 0 \\ -1 \\ -1 \\ 0 \\ 0 -\end{bmatrix} +0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 +\end{bmatrix}^{T} \end{equation} diff --git a/latex/Questions/Q16.tex b/latex/Questions/Q16.tex index a4038e6..4310d46 100644 --- a/latex/Questions/Q16.tex +++ b/latex/Questions/Q16.tex @@ -3,7 +3,7 @@ La Figure~\ref{fig:q16_perturbation} montre que le système ne corrige pas l'erreur due à la perturbation. \begin{figure}[H] \centering - \includegraphics[width=0.4\textwidth]{Illustrations/im3} % Remplacez par votre image + \includegraphics[scale=0.3]{Illustrations/im3} % Remplacez par votre image \caption{Simulation du système avec une perturbation.} \label{fig:q16_perturbation} \end{figure} diff --git a/latex/Questions/Q17.log b/latex/Questions/Q17.log new file mode 100644 index 0000000..80b4dac --- /dev/null +++ b/latex/Questions/Q17.log @@ -0,0 +1,413 @@ +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 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**Q17.tex +(./Q17.tex +LaTeX2e <2023-11-01> patch level 1 +L3 programming layer <2024-01-22> + +! 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Emergency stop. +<*> Q17.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 37 strings out of 474222 + 630 string characters out of 5748733 + 1923975 words of memory out of 5000000 + 22390 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 +! ==> Fatal error occurred, no output PDF file produced! diff --git a/latex/Questions/Q17.tex b/latex/Questions/Q17.tex index 6234226..8674fbf 100644 --- a/latex/Questions/Q17.tex +++ b/latex/Questions/Q17.tex @@ -2,19 +2,34 @@ À partir des huit pôles obtenus avec la commande par retour d'état de la Question 11, on ajoute un pôle supplémentaire associé à l'action intégrale $p_i = -4{,}3805$, correspondant au pôle du système le plus rapide. Ainsi, les pôles desirés du système augmenté sont : +\begin{center} +$p_{\text{des}} =$ +\begin{equation} +\begin{bmatrix} +-85 & +-8+69i & +-8-69i & +-28+17i & +-28-17i & +-16 & +-4+3i & +-4-3i & +-4 +\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}. +-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/Q18.tex b/latex/Questions/Q18.tex index 6243f36..df69cb9 100644 --- a/latex/Questions/Q18.tex +++ b/latex/Questions/Q18.tex @@ -9,12 +9,12 @@ K_1 & K_i \begin{equation} K_1 = \begin{bmatrix} --3{,}1081 & -18{,}9367 & -0{,}1707 & 2{,}1192 & 19{,}4677 & -2{,}3049 & 2{,}3377 & -1{,}7524 +-3.1081 & -18.9367 & -0.1707 & 2.1192 & 19.4677 & -2.3049 & 2.3377 & -1.7524 \end{bmatrix} \end{equation} \begin{equation} K_i = \begin{bmatrix} -26{,}2831 +26.2831 \end{bmatrix} \end{equation} diff --git a/latex/Questions/Q5.log b/latex/Questions/Q5.log new file mode 100644 index 0000000..998b70c --- /dev/null +++ b/latex/Questions/Q5.log @@ -0,0 +1,335 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.4.14) 7 MAY 2026 19:15 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**Q5.tex +(./Q5.tex +LaTeX2e <2023-11-01> patch level 1 +L3 programming layer <2024-01-22> + +! 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Emergency stop. +<*> Q5.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 36 strings out of 474222 + 610 string characters out of 5748733 + 1922975 words of memory out of 5000000 + 22389 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,1n,32p,169b,91s stack positions out of 10000i,1000n,20000p,200000b,200000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/latex/Questions/Q5.tex b/latex/Questions/Q5.tex index cc21bac..3b8788e 100644 --- a/latex/Questions/Q5.tex +++ b/latex/Questions/Q5.tex @@ -5,6 +5,6 @@ La Figure 2 montre que la sortie $y(t)$ oscille après l’échelon, confirmant \begin{figure}[H] \centering - \includegraphics[scale=0.4]{Illustrations/Q5stepReponse} % Remplacez par votre image + \includegraphics[scale=0.3]{Illustrations/Q5stepReponse} % Remplacez par votre image \caption{Response du système en boucle ouverte.} \end{figure} \ No newline at end of file diff --git a/latex/Questions/Q7.tex b/latex/Questions/Q7.tex index de72eeb..45449d4 100644 --- a/latex/Questions/Q7.tex +++ b/latex/Questions/Q7.tex @@ -9,4 +9,4 @@ Le gain $H$ est ensuite calculé à partir d'un système en boucle fermée sous $H=-\frac{1}{C(A-BKC)^{-1}B}$ \end{center} Ainsi, les valeurs retenues pour la commande en boucle fermée sont : -$K = 1{,}85$ \textit{\&} $H = -3$ +$\textbf{K = 1.85}$ \textit{\&} $\textbf{H = -3}$ diff --git a/latex/Questions/Q8.tex b/latex/Questions/Q8.tex index 61f3135..faa82e6 100644 --- a/latex/Questions/Q8.tex +++ b/latex/Questions/Q8.tex @@ -4,7 +4,7 @@ La Figure 3 montre que $w(L,t)$ suit la consigne $w_c(t)$ unitaire, avec une err \begin{figure}[H] \centering - \includegraphics[scale=0.4]{Illustrations/Q8} % Remplacez par votre image + \includegraphics[scale=0.3]{Illustrations/Q8} % Remplacez par votre image \caption{Suivi de la consigne $w_c(t)$ par la position $w(L,t)$.} \label{fig:Q8_1} \end{figure} \ No newline at end of file diff --git a/latex/Questions/Q9.tex b/latex/Questions/Q9.tex index d55442a..c413731 100644 --- a/latex/Questions/Q9.tex +++ b/latex/Questions/Q9.tex @@ -1,39 +1,34 @@ \textbf{Question 9} -La Figure~\ref{fig:q9_eigenvalues} compare les valeurs propres du système en boucle fermée en temps continu avec celles obtenues après discrétisation par ZOH, pour les trois périodes d'échantillonnage proposées. +La Figure~\ref{fig:q9_eigenvalues} compare les valeurs propres du système en boucle fermée en temps continu avec celles obtenues après discrétisation par ZOH, pour les trois périodes d'échantillonnage proposées: $T_e = \{0.01s ; 0.02s ; 0.04s\}$. - -\begin{figure}[h] - \centering +\begin{figure}[H] +\centering \begin{subfigure}{0.48\textwidth} \centering - \includegraphics[width=\textwidth]{Illustrations/Q9continu} + \includegraphics[scale=0.3]{Illustrations/Q9continu} \caption{Valeurs propres en temps continu.} \end{subfigure} \hfill \begin{subfigure}{0.48\textwidth} \centering - \includegraphics[width=\textwidth]{Illustrations/Q9001} - \caption{$T_s = 0{,}01\,\mathrm{s}$.} + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q9_TE001} + \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/Q9002} - \caption{$T_s = 0{,}02\,\mathrm{s}$.} + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q9_TE002} + \caption{$T_s = 0.02\,\mathrm{s}$.} \end{subfigure} \hfill \begin{subfigure}{0.48\textwidth} \centering - \includegraphics[width=\textwidth]{Illustrations/Q9004} - \caption{$T_s = 0{,}04\,\mathrm{s}$.} + \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q9_TE004} + \caption{$T_s = 0.04\,\mathrm{s}$.} \end{subfigure} - \caption{Comparaison des valeurs propres du système en boucle fermée.} \label{fig:q9_eigenvalues} \end{figure} -Les pôles discrets restent à l'intérieur du cercle unité pour les trois valeurs de $T_s$, ce qui montre que le système discret reste stable. +Comme nous pouvons voir, les pôles du système discret restent à l'interieur du cercle unitaire, mais dès que nous passons à $T_e=0.04s$ ça rend le système instable avec un pôle hors le cercle. diff --git a/latex/contents.tex b/latex/contents.tex index 4ce5f09..d82cc35 100644 --- a/latex/contents.tex +++ b/latex/contents.tex @@ -9,7 +9,6 @@ \subsection{Discrétisation par éléments finis} \input{Questions/Q2} \input{Questions/Q3} -\newpage \input{Questions/Q4} \input{Questions/Q5} \input{Questions/Q6} @@ -21,10 +20,7 @@ \input{Questions/Q8} \subsection{Retour de sortie par ZOH} -%%newpage pour les imgs q9 -\newpage \input{Questions/Q9} -\newpage \input{Questions/Q10} diff --git a/latex/main.aux b/latex/main.aux index 5fa9b14..dcadd29 100644 --- a/latex/main.aux +++ b/latex/main.aux @@ -15,32 +15,27 @@ \@writefile{toc}{\contentsline {subsection}{\numberline 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