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Changement de la matrice B

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Tournesol02 2026-05-07 19:58:41 +02:00
parent eb8f8d1cdd
commit 341b653101
9 changed files with 217 additions and 139 deletions
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2
latex/Questions/Q13.tex
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@ -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}

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@ -1,4 +1,4 @@
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@ -339,30 +339,9 @@ Overfull \hbox (4.44444pt too wide) in paragraph at lines 3--5
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See the LaTeX manual or LaTeX Companion for explanation.
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l.21 p
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! Missing $ inserted.
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l.21 p_
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l.6 $p_{\text
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@ -370,31 +349,128 @@ spelling (e.g., `I\hbox'). Otherwise just continue,
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l.22 \begin{bmatrix}
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l.9 -85 &
I can't figure out why you would want to use a tab mark
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l.11 -8-69i &
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I can't figure out why you would want to use a tab mark
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I can't figure out why you would want to use a tab mark
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I can't figure out why you would want to use a tab mark
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latex/Questions/Q17.tex
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@ -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}} =
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%-4.3805
%\end{bmatrix}
%\end{equation}
\end{center}

18
latex/Questions/Q3.tex
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@ -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}

31
latex/Questions/Q4.tex
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@ -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.

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