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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
\/cmr/m/n/10 ^^Se
[]
! LaTeX Error: Missing \begin{document}.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.21 p
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Missing character: There is no p in font nullfont!
! Missing $ inserted.
<inserted text>
$
l.21 p_
{\text{des}} =
I've inserted a begin-math/end-math symbol since I think
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l.6 $p_{\text
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The control sequence at the end of the top line
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@ -370,31 +349,128 @@ spelling (e.g., `I\hbox'). Otherwise just continue,
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! LaTeX Error: Environment bmatrix undefined.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.22 \begin{bmatrix}
l.8 \begin{bmatrix}
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Type I <command> <return> to replace it with another command,
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! Misplaced alignment tab character &.
l.9 -85 &
! LaTeX Error: \begin{document} ended by \end{bmatrix}.
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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l.11 -8-69i &
I can't figure out why you would want to use a tab mark
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l.12 -28+17i &
I can't figure out why you would want to use a tab mark
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l.13 -28-17i &
I can't figure out why you would want to use a tab mark
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l.15 -4+3i &
I can't figure out why you would want to use a tab mark
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l.16 -4-3i &
I can't figure out why you would want to use a tab mark
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30
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}

27
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.
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]
\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}
\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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