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mise a jour Q-11-12-13

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\textbf{Question 10}
Nous avons obtenu une valeur maximale de $0{,}5\,\mathrm{s}$ pour la période d'échantillonnage.
Manque l'image!!!!!!!!!!!

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\textbf{Question 11}
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
\end{bmatrix}
\end{equation}

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\textbf{Question 12}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{Illustrations/Q12} % Remplacez par votre image
\caption{Simulation numérique du système continu en boucle fermée.}
\end{figure}

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\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}.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{Illustrations/im1} % Remplacez par votre image
\caption{Valeurs propres pour différentes périodes d'échantillonnage.}
\label{fig:q13}
\end{figure}

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\newpage
\textbf{Question 9}
REWRITE
\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.
\begin{figure}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
%%\includegraphics[width=\textwidth]{Illustrations/Q9Poles}
\includegraphics[width=\textwidth]{Illustrations/Q9continu}
\caption{Valeurs propres en temps continu.}
\end{subfigure}
\hfill
\begin{subfigure}{0.48\textwidth}
\centering
%%\includegraphics[width=\textwidth]{Illustrations/Q9001}
\includegraphics[width=\textwidth]{Illustrations/Q9001}
\caption{$T_s = 0{,}01\,\mathrm{s}$.}
\end{subfigure}
@ -22,15 +21,18 @@ REWRITE
\begin{subfigure}{0.48\textwidth}
\centering
%%\includegraphics[width=\textwidth]{Illustrations/Q9002_}
\includegraphics[width=\textwidth]{Illustrations/Q9002}
\caption{$T_s = 0{,}02\,\mathrm{s}$.}
\end{subfigure}
\hfill
\begin{subfigure}{0.48\textwidth}
\centering
%%\includegraphics[width=\textwidth]{Illustrations/Q9004_}
\includegraphics[width=\textwidth]{Illustrations/Q9004}
\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.

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Une introduction
\newpage
\section{Dynamique du système avec équations différentielles partielles}
\subsection{Espace d'état}
\section{Modélisation du système}
%%%Q1
\input{Questions/Q1}
%%%
\subsection{Discrétisation par éléments finis}
%%Q2
\input{Questions/Q2}
%%%%
@ -21,15 +19,27 @@ Une introduction
\input{Questions/Q4}
%%
%%%Q5
\newpage
\input{Questions/Q5}
%%Q6
\section{Retour de sortie}
%%
\input{Questions/Q7}
\input{Questions/Q8}
%%newpage pour les imgs q9
\newpage
\input{Questions/Q9}
\input{Questions/Q10}
% \section{Une autre section}
\section{Retour d'état}
\input{Questions/Q11}
\input{Questions/Q12}
\subsection{Retour d'état par ZOH}
\input{Questions/Q13}
\input{Questions/Q14}
% \begin{table}[h]

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