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19
latex/Questions/Q10.tex
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@ -1,12 +1,21 @@
\textbf{Question 10} \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] \begin{figure}[H]
\centering \centering
\includegraphics[scale=0.4]{Illustrations/Q10} \begin{subfigure}{0.48\textwidth}
\caption{Système à $T_s = 2s$} \centering
\label{fig:Q10_1} \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} \end{figure}

2
latex/Questions/Q11.tex
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@ -5,6 +5,6 @@ Avec la fonction \texttt{lqr} de MATLAB, le gain de retour d'état obtenu est :
\begin{equation} \begin{equation}
K = K =
\begin{bmatrix} \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{bmatrix}
\end{equation} \end{equation}

10
latex/Questions/Q13.tex
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@ -1,12 +1,14 @@
\textbf{Question 13} \textbf{Question 13}
Nous avons testé plusieurs périodes d'échantillonnage entre $0{,}1\,\mathrm{s}$ et $5\,\mathrm{s}$. Avec le même periode d'echantillonnage comme dans la section precedente: $T_e=0.0356$ notre système est rendu instable.
Les valeurs propres discrètes restent à l'intérieur du cercle unité, comme le montre la Figure~\ref{fig:q13}.
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[width=0.4\textwidth]{Illustrations/im1} % Remplacez par votre image \includegraphics[scale=0.3]{Illustrations/NouveauxImages/Q13}
\caption{Valeurs propres pour différentes périodes d'échantillonnage.} \caption{Pôles du système avec temps d'echantillonnage: $T_e = 0.0356$}
\label{fig:q13} \label{fig:q13}
\end{figure} \end{figure}

29
latex/Questions/Q14.tex
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\textbf{Question 14} \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] \begin{figure}[H]
\centering \centering
\includegraphics[width=0.4\textwidth]{Illustrations/Q14} % Remplacez par votre image \begin{subfigure}{0.48\textwidth}
\caption{Réponse du système à $T_s = 0.3s$.} \centering
\label{fig:q14_reponses} \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} \end{figure}

4
latex/Questions/Q15.tex
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@ -7,7 +7,7 @@ B_p =
\end{bmatrix} \end{bmatrix}
= =
\begin{bmatrix} \begin{bmatrix}
0 \\ 0 \\ 0 \\ 0 \\ -1 \\ -1 \\ 0 \\ 0 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0
\end{bmatrix} \end{bmatrix}^{T}
\end{equation} \end{equation}

2
latex/Questions/Q16.tex
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@ -3,7 +3,7 @@
La Figure~\ref{fig:q16_perturbation} montre que le système ne corrige pas l'erreur due à la perturbation. La Figure~\ref{fig:q16_perturbation} montre que le système ne corrige pas l'erreur due à la perturbation.
\begin{figure}[H] \begin{figure}[H]
\centering \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.} \caption{Simulation du système avec une perturbation.}
\label{fig:q16_perturbation} \label{fig:q16_perturbation}
\end{figure} \end{figure}

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l.21 p
_{\text{des}} =
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l.22 \begin{bmatrix}
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l.32 \end{bmatrix}
^{T}
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latex/Questions/Q17.tex
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@ -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. 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 : 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} \begin{equation}
p_{\text{des}} = p_{\text{des}} =
\begin{bmatrix} \begin{bmatrix}
-84{,}6647\\ -84.6647 \\
-7{,}5585+68{,}7669i\\ -7.5585+68.7669i \\
-7{,}5585-68{,}7669i\\ -7.5585-68.7669i \\
-27{,}9691+17{,}2408i\\ -27.9691+17.2408i \\
-27{,}9691-17{,}2408i\\ -27.9691-17.2408i \\
-15{,}9108\\ -15.9108 \\
-4{,}3805+2{,}6626i\\ -4.3805+2.6626i \\
-4{,}3805-2{,}6626i\\ -4.3805-2.6626i \\
-4{,}3805 -4.3805
\end{bmatrix}. \end{bmatrix}
\end{equation} \end{equation}
\end{center}

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latex/Questions/Q18.tex
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@ -9,12 +9,12 @@ K_1 & K_i
\begin{equation} \begin{equation}
K_1 = K_1 =
\begin{bmatrix} \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{bmatrix}
\end{equation} \end{equation}
\begin{equation} \begin{equation}
K_i = K_i =
\begin{bmatrix} \begin{bmatrix}
26{,}2831 26.2831
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}

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a Figure 2 montre que la sortie $y(t)$ oscille après l’échelon, con...
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l.4 ...2 montre que la sortie $y(t)$ oscille aprè
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2
latex/Questions/Q5.tex
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@ -5,6 +5,6 @@ La Figure 2 montre que la sortie $y(t)$ oscille après l’échelon, confirmant
\begin{figure}[H] \begin{figure}[H]
\centering \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.} \caption{Response du système en boucle ouverte.}
\end{figure} \end{figure}

2
latex/Questions/Q7.tex
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@ -9,4 +9,4 @@ Le gain $H$ est ensuite calculé à partir d'un système en boucle fermée sous
$H=-\frac{1}{C(A-BKC)^{-1}B}$ $H=-\frac{1}{C(A-BKC)^{-1}B}$
\end{center} \end{center}
Ainsi, les valeurs retenues pour la commande en boucle fermée sont : Ainsi, les valeurs retenues pour la commande en boucle fermée sont :
$K = 1{,}85$ \textit{\&} $H = -3$ $\textbf{K = 1.85}$ \textit{\&} $\textbf{H = -3}$

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latex/Questions/Q8.tex
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@ -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] \begin{figure}[H]
\centering \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 $w(L,t)$.} \caption{Suivi de la consigne $w_c(t)$ par la position $w(L,t)$.}
\label{fig:Q8_1} \label{fig:Q8_1}
\end{figure} \end{figure}

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latex/Questions/Q9.tex
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@ -1,39 +1,34 @@
\textbf{Question 9} \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]
\begin{figure}[h] \centering
\centering
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\centering \centering
\includegraphics[width=\textwidth]{Illustrations/Q9continu} \includegraphics[scale=0.3]{Illustrations/Q9continu}
\caption{Valeurs propres en temps continu.} \caption{Valeurs propres en temps continu.}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\centering \centering
\includegraphics[width=\textwidth]{Illustrations/Q9001} \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q9_TE001}
\caption{$T_s = 0{,}01\,\mathrm{s}$.} \caption{$T_s = 0.01\,\mathrm{s}$.}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.48\textwidth}
\vspace{0.3cm}
\begin{subfigure}{0.48\textwidth}
\centering \centering
\includegraphics[width=\textwidth]{Illustrations/Q9002} \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q9_TE002}
\caption{$T_s = 0{,}02\,\mathrm{s}$.} \caption{$T_s = 0.02\,\mathrm{s}$.}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\centering \centering
\includegraphics[width=\textwidth]{Illustrations/Q9004} \includegraphics[scale=0.4]{Illustrations/NouveauxImages/Q9_TE004}
\caption{$T_s = 0{,}04\,\mathrm{s}$.} \caption{$T_s = 0.04\,\mathrm{s}$.}
\end{subfigure} \end{subfigure}
\caption{Comparaison des valeurs propres du système en boucle fermée.}
\label{fig:q9_eigenvalues} \label{fig:q9_eigenvalues}
\end{figure} \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.

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latex/contents.tex
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@ -9,7 +9,6 @@
\subsection{Discrétisation par éléments finis} \subsection{Discrétisation par éléments finis}
\input{Questions/Q2} \input{Questions/Q2}
\input{Questions/Q3} \input{Questions/Q3}
\newpage
\input{Questions/Q4} \input{Questions/Q4}
\input{Questions/Q5} \input{Questions/Q5}
\input{Questions/Q6} \input{Questions/Q6}
@ -21,10 +20,7 @@
\input{Questions/Q8} \input{Questions/Q8}
\subsection{Retour de sortie par ZOH} \subsection{Retour de sortie par ZOH}
%%newpage pour les imgs q9
\newpage
\input{Questions/Q9} \input{Questions/Q9}
\newpage
\input{Questions/Q10} \input{Questions/Q10}

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{\reset@font\stcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode \numberline {2.1}Retour de sortie par ZOH}{\reset@font\stcSSfont 5}{subsection.2.1}} {\reset@font\stcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode \numberline {2.1}Retour de sortie par ZOH}{\reset@font\stcSSfont 4}{subsection.2.1}}

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{\reset@font\stcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode \numberline {3.1}Retour d'état par ZOH}{\reset@font\stcSSfont 7}{subsection.3.1}} {\reset@font\stcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode \numberline {3.1}Retour d'état par ZOH}{\reset@font\stcSSfont 6}{subsection.3.1}}

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{\reset@font\stcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode \numberline {4.1}Remplacement du pré-gain H}{\reset@font\stcSSfont 9}{subsection.4.1}} {\reset@font\stcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode \numberline {4.1}Remplacement du pré-gain H}{\reset@font\stcSSfont 7}{subsection.4.1}}

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\parbox{\textwidth}{ \parbox{\textwidth}{
\sffamily % arial \sffamily % arial
\textbf{Brage JOHNSEN}\\ \textbf{Brage JOHNSEN}\\
\textbf{Gabriela DONZELLI}\\ \textbf{Gabriela DONZELLI}\\
\textbf{Thibaut ALETRUT}\\ \textbf{Thibaut ALETRUT}\\
Département GEI\\ Département GEI\\
4A-AE\\ 4A-AE\\

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\contentsline {subsection}{\numberline {1.1}Dynamique avec EDP}{1}{subsection.1.1}% \contentsline {subsection}{\numberline {1.1}Dynamique avec EDP}{1}{subsection.1.1}%
\contentsline {subsection}{\numberline {1.2}Discrétisation par éléments finis}{1}{subsection.1.2}% \contentsline {subsection}{\numberline {1.2}Discrétisation par éléments finis}{1}{subsection.1.2}%
\contentsline {section}{\numberline {2}Retour de sortie}{4}{section.2}% \contentsline {section}{\numberline {2}Retour de sortie}{4}{section.2}%
\contentsline {subsection}{\numberline {2.1}Retour de sortie par ZOH}{5}{subsection.2.1}% \contentsline {subsection}{\numberline {2.1}Retour de sortie par ZOH}{4}{subsection.2.1}%
\contentsline {section}{\numberline {3}Retour d'état}{7}{section.3}% \contentsline {section}{\numberline {3}Retour d'état}{5}{section.3}%
\contentsline {subsection}{\numberline {3.1}Retour d'état par ZOH}{7}{subsection.3.1}% \contentsline {subsection}{\numberline {3.1}Retour d'état par ZOH}{6}{subsection.3.1}%
\contentsline {section}{\numberline {4}Rejet de perturbation}{8}{section.4}% \contentsline {section}{\numberline {4}Rejet de perturbation}{7}{section.4}%
\contentsline {subsection}{\numberline {4.1}Remplacement du pré-gain H}{9}{subsection.4.1}% \contentsline {subsection}{\numberline {4.1}Remplacement du pré-gain H}{7}{subsection.4.1}%