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sous cette forme: \begin{equation} \dot{x} = diff --git a/latex/Questions/Q10.tex b/latex/Questions/Q10.tex index b524c7f..09933ab 100644 --- a/latex/Questions/Q10.tex +++ b/latex/Questions/Q10.tex @@ -1,7 +1,12 @@ \textbf{Question 10} -Nous avons obtenu une valeur maximale de $0{,}5\,\mathrm{s}$ pour la période d'échantillonnage. +Nous avons trouvé un seuil de $T_s = 0.0356s$ à l'aide d'un boucle qui traverse les +\begin{figure}[H] + \centering + \includegraphics[scale=0.4]{Illustrations/Q10} + \caption{Système à $T_s = 2s$} + \label{fig:Q10_1} +\end{figure} -Manque l'image!!!!!!!!!!! diff --git a/latex/Questions/Q12.tex b/latex/Questions/Q12.tex index 3d4de16..1d96eeb 100644 --- a/latex/Questions/Q12.tex +++ b/latex/Questions/Q12.tex @@ -1,6 +1,6 @@ \textbf{Question 12} -La Figure~\ref{fig:Q15} montre que le système en boucle fermée suit correctement la référence et converge vers la consigne en régime permanent. +La Figure~\ref{fig:Q15} montre que le système en boucle fermée suit correctement la référence avec un léger depassement et converge vers la consigne en régime permanent. \begin{figure}[H] \centering diff --git a/latex/Questions/Q14.tex b/latex/Questions/Q14.tex index 3dc8359..ca82350 100644 --- a/latex/Questions/Q14.tex +++ b/latex/Questions/Q14.tex @@ -1,11 +1,10 @@ \textbf{Question 14} -D'après la Figure~\ref{fig:q14_reponses}, les petites périodes d'échantillonnage permettent de mieux approcher la réponse temporelle continue, comme le montre la Figure~\ref{fig:Q15}. -On choisit donc $T_s = 0{,}01\,\mathrm{s}$, car cette valeur conserve un comportement proche de la réponse continue. +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}. \begin{figure}[H] \centering - \includegraphics[width=0.4\textwidth]{Illustrations/im2} % Remplacez par votre image - \caption{Réponses temporelles du système pour différentes périodes d'échantillonnage.} + \includegraphics[width=0.4\textwidth]{Illustrations/Q14} % Remplacez par votre image + \caption{Réponse du système à $T_s = 0.3s$.} \label{fig:q14_reponses} \end{figure} \ No newline at end of file diff --git a/latex/Questions/Q15.tex b/latex/Questions/Q15.tex index a13f61e..ccb0a3e 100644 --- a/latex/Questions/Q15.tex +++ b/latex/Questions/Q15.tex @@ -1,4 +1,13 @@ \textbf{Question 15} - -:o +Après avoir calculé, nous retrouvons la matrice $B_p$ sous la forme: +\begin{equation} +B_p = +\begin{bmatrix} +0 \\ -E^{-1}\int_0^L \Phi(\zeta)d\zeta +\end{bmatrix} += +\begin{bmatrix} +0 \\ 0 \\ 0 \\ 0 \\ -1 \\ -1 \\ 0 \\ 0 +\end{bmatrix} +\end{equation} diff --git a/latex/Questions/Q16.tex b/latex/Questions/Q16.tex index 0a6f821..a4038e6 100644 --- a/latex/Questions/Q16.tex +++ b/latex/Questions/Q16.tex @@ -1,6 +1,5 @@ \textbf{Question 16} - La Figure~\ref{fig:q16_perturbation} montre que le système ne corrige pas l'erreur due à la perturbation. \begin{figure}[H] \centering diff --git a/latex/Questions/Q17.tex b/latex/Questions/Q17.tex index 1486807..6234226 100644 --- a/latex/Questions/Q17.tex +++ b/latex/Questions/Q17.tex @@ -1,15 +1,10 @@ \textbf{Question 17} -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. +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{equation} -p_i = -4{,}3805 -\end{equation} - -Ainsi, les pôles désirés du système augmenté sont : - -\begin{equation} -p_{\text{dés}} = +p_{\text{des}} = \begin{bmatrix} -84{,}6647\\ -7{,}5585+68{,}7669i\\ diff --git a/latex/Questions/Q18.tex b/latex/Questions/Q18.tex index 0d4f986..6243f36 100644 --- a/latex/Questions/Q18.tex +++ b/latex/Questions/Q18.tex @@ -1,17 +1,20 @@ \textbf{Question 18} -En utilisant la fonction \texttt{place} sous MATLAB, nous avons calculé les gains $K_1$ et $K_i$ : - +En utilisant la fonction \texttt{place} sous MATLAB, nous avons calculé les gains $K_1$ et $K_i$ à partir du système augmenté: +\begin{equation} +K_{Aug} = \begin{bmatrix} +K_1 & K_i +\end{bmatrix} +\end{equation} \begin{equation} K_1 = \begin{bmatrix} -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 -\end{bmatrix}. +\end{bmatrix} \end{equation} diff --git a/latex/Questions/Q19.tex b/latex/Questions/Q19.tex index 7f66f0a..e12b8cf 100644 --- a/latex/Questions/Q19.tex +++ b/latex/Questions/Q19.tex @@ -1,6 +1,6 @@ \textbf{Question 19} -La Figure~\ref{fig:q19_integrale} montre qu'avec l'action intégrale, la perturbation est rejetée et l'erreur en régime permanent est annulée. +La Figure~\ref{fig:q19_integrale} montre qu'avec l'action intégrale, la perturbation est rejetée et l'erreur en régime permanent est corrigée. \begin{figure}[H] \centering diff --git a/latex/Questions/Q2.log b/latex/Questions/Q2.log new file mode 100644 index 0000000..3bd1068 --- /dev/null +++ b/latex/Questions/Q2.log @@ -0,0 +1,890 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.4.14) 7 MAY 2026 11:53 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**Q2.tex +(./Q2.tex +LaTeX2e <2023-11-01> patch level 1 +L3 programming layer <2024-01-22> + +! 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.25 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 25--25 +[] + [] + + +Overfull \hbox (180.4728pt too wide) detected at line 33 +\OML/cmm/m/it/10 ^^^\/cmr/m/n/10 (\OML/cmm/m/it/10 ^^P\/cmr/m/n/10 )[] [] = \OM +L/cmm/m/it/10 ^^^\/cmr/m/n/10 (\OML/cmm/m/it/10 ^^P\/cmr/m/n/10 )\OML/cmm/m/it/ +10 ^^^\/cmr/m/n/10 (\OML/cmm/m/it/10 ^^P\/cmr/m/n/10 )[][](\OML/cmm/m/it/10 t\/ +cmr/m/n/10 ) + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.35 C + omme $e_{2d}(t)$ ne dépend pas de $\zeta$, il peut être sorti de l'i... + +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.49 A + près avoir fait une integration par partie: +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.53 O + n identifie alors les matrices $E$ et $D$. Ainsi, les deux équations ... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 53--54 +[] + [] + + +Overfull \hbox (7.95831pt too wide) in paragraph at lines 53--54 +\OML/cmm/m/it/10 E$ + [] + + +Overfull \hbox (8.55695pt too wide) in paragraph at lines 53--54 +\OML/cmm/m/it/10 D$ + [] + + +Overfull \hbox (4.44444pt too wide) in paragraph at lines 53--54 +\/cmr/m/n/10 ^^Se + [] + + +Overfull \hbox (4.44444pt too wide) in paragraph at lines 53--54 +\/cmr/m/n/10 ^^Se + [] + + +Overfull \hbox (8.88889pt too wide) in paragraph at lines 53--54 +\/cmr/m/n/10 ^^Se^^Se + [] + + +Overfull \hbox (4.44444pt too wide) in paragraph at lines 53--54 +\/cmr/m/n/10 ^^Se + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.55 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 55--55 +[] + [] + + +! LaTeX Error: Environment aligned undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.56 \begin{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.57 E\dot{x}_{1d}(t) & + = D e_{2d}(t),\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.58 E\dot{x}_{2d}(t) & + = -D^T e_{1d}(t)-\phi(L)u(t)-F_{ext}. +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 55 ended by \end{aligned}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.59 \end{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (253.38242pt too wide) detected at line 60 +\OML/cmm/m/it/10 E[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = \OML/cmm/m +/it/10 De[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] +E[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = \OMS/cmsy/m/n/10 ^^@\OML/cm +m/m/it/10 D[]e[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) \OMS/cmsy/m/n/10 + ^^@ \OML/cmm/m/it/10 ^^^\/cmr/m/n/10 (\OML/cmm/m/it/10 L\/cmr/m/n/10 )\OML/cmm +/m/it/10 u\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \ +OML/cmm/m/it/10 F[]: + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.62 E + nfin, on obtient : +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 62--63 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.64 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 64--64 +[] + [] + +! Undefined control sequence. +l.65 \boxed + { +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Environment aligned undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.66 \begin{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.67 E\dot{x}_{1d}(t) & + = D e_{2d}(t),\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.68 E\dot{x}_{2d}(t) & + = -D^T e_{1d}(t)-\phi(L)u(t)-F_{ext},\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.69 y(t) & + = -\phi(L)^T e_{2d}(t). +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 64 ended by \end{aligned}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.70 \end{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (356.01259pt too wide) detected at line 72 +[] + [] + +) +! Emergency stop. +<*> Q2.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 41 strings out of 474222 + 714 string characters out of 5748733 + 1925975 words of memory out of 5000000 + 22394 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,163b,91s stack positions out of 10000i,1000n,20000p,200000b,200000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/latex/Questions/Q2.tex b/latex/Questions/Q2.tex index 1a5e35c..c3a1667 100644 --- a/latex/Questions/Q2.tex +++ b/latex/Questions/Q2.tex @@ -46,6 +46,11 @@ Comme $e_{2d}(t)$ ne dépend pas de $\zeta$, il peut être sorti de l'intégrale e_{2d}(t) \end{equation} +Après avoir fait une integration par partie: +à inserer:\\ + + + On identifie alors les matrices $E$ et $D$. Ainsi, les deux équations d'état discrétisées s'écrivent : \begin{equation} diff --git a/latex/Questions/Q20.tex b/latex/Questions/Q20.tex index e69de29..adc008a 100644 --- a/latex/Questions/Q20.tex +++ b/latex/Questions/Q20.tex @@ -0,0 +1,3 @@ +\textbf{Question 20} + +Nous avons testé avec 6 decimals, mais la valeur absolue d'un des pôles z: $|z|>1$, alors nous n'arrivons jamais à stabiliser le système. Aussi, nous considerons avec $T_s \rightarrow 0.00001$ est un temps continu, la commande n'est pas implemantable sur un controleur numérique. \ No newline at end of file diff --git a/latex/Questions/Q3.log b/latex/Questions/Q3.log new file mode 100644 index 0000000..7876b4e --- /dev/null +++ b/latex/Questions/Q3.log @@ -0,0 +1,1406 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.4.14) 7 MAY 2026 12:03 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**Q3.tex +(./Q3.tex +LaTeX2e <2023-11-01> patch level 1 +L3 programming layer <2024-01-22> + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \textbf{Question 3} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (64.64003pt too wide) in paragraph at lines 1--2 +[]\OT1/cmr/bx/n/10 Question + [] + + +Overfull \hbox (5.74997pt too wide) in paragraph at lines 1--2 +\OT1/cmr/bx/n/10 3 + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3 L + e système devient avec $\rho = EI = 1$ et $q(\zeta,t)=0$ : +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +LaTeX Font Info: Trying to load font information for +cmr on input line 3. +LaTeX Font Info: No file cmr.fd. on input line 3. + +LaTeX Font Warning: Font shape `/cmr/m/n' undefined +(Font) using `/cmr/m/n' instead on input line 3. + +! Corrupted NFSS tables. +wrong@fontshape ...message {Corrupted NFSS tables} + error@fontshape else let f... +l.3 Le systè + me devient avec $\rho = EI = 1$ et $q(\zeta,t)=0$ : +This error message was generated by an \errmessage +command, so I can't give any explicit help. +Pretend that you're Hercule Poirot: Examine all clues, +and deduce the truth by order and method. + + +LaTeX Font Warning: Font shape `/cmr/m/n' undefined +(Font) using `OT1/cmr/m/n' instead on input line 3. + +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 3. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 3. +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 3--4 +[] + [] + + +Overfull \hbox (9.44446pt too wide) in paragraph at lines 3--4 +\/cmr/m/n/10 ^^Re + [] + + +Overfull \hbox (15.72566pt too wide) in paragraph at lines 3--4 +\OML/cmm/m/it/10 ^^Z \/cmr/m/n/10 = + [] + + +Overfull \hbox (23.69437pt too wide) in paragraph at lines 3--4 +\OML/cmm/m/it/10 EI \/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 3--4 +\/cmr/m/n/10 1$ + [] + + +Overfull \hbox (36.3246pt too wide) in paragraph at lines 3--4 +\OML/cmm/m/it/10 q\/cmr/m/n/10 (\OML/cmm/m/it/10 ^^P; t\/cmr/m/n/10 ) = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 3--4 +\/cmr/m/n/10 0$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.5 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 5--5 +[] + [] + + +! LaTeX Error: Environment aligned undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.6 \begin{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.7 E\dot{x}_{1d}(t) & + = D x_{2d}(t),\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.8 E\dot{x}_{2d}(t) & + = -D^T x_{1d}(t)-\phi(L)u(t),\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.9 y(t) & + = -\phi(L)^T x_{2d}(t). +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 5 ended by \end{aligned}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.10 \end{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (324.24289pt too wide) detected at line 11 +\OML/cmm/m/it/10 E[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = \OML/cmm/m +/it/10 Dx[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] +E[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = \OMS/cmsy/m/n/10 ^^@\OML/cm +m/m/it/10 D[]x[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) \OMS/cmsy/m/n/10 + ^^@ \OML/cmm/m/it/10 ^^^\/cmr/m/n/10 (\OML/cmm/m/it/10 L\/cmr/m/n/10 )\OML/cmm +/m/it/10 u\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] y +\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = \OMS/cmsy/m/n/10 ^^@\OML/cmm/m +/it/10 ^^^\/cmr/m/n/10 (\OML/cmm/m/it/10 L\/cmr/m/n/10 )[]\OML/cmm/m/it/10 x[]\ +/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )\OML/cmm/m/it/10 : + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.13 E + n multipliant les deux premières équations par $E^{-1}$, on obtient : +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 13--14 +[] + [] + + +Overfull \hbox (4.44444pt too wide) in paragraph at lines 13--14 +\/cmr/m/n/10 ^^Re + [] + + +Overfull \hbox (4.44444pt too wide) in paragraph at lines 13--14 +\/cmr/m/n/10 ^^Se + [] + + +Overfull \hbox (18.69446pt too wide) in paragraph at lines 13--14 +\OML/cmm/m/it/10 E[]$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.15 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 15--15 +[] + [] + + +! LaTeX Error: Environment aligned undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.16 \begin{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.17 \dot{x}_{1d}(t) & + = E^{-1}D x_{2d}(t),\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.18 \dot{x}_{2d}(t) & + = -E^{-1}D^T x_{1d}(t)-E^{-1}\phi(L)u(t). +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 15 ended by \end{aligned}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.19 \end{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (269.60913pt too wide) detected at line 20 +[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = \OML/cmm/m/it/10 E[]Dx[]\/cm +r/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] []\/cmr/m/n/10 +(\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = \OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 E[]D[] +x[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \OML/cmm +/m/it/10 E[]^^^\/cmr/m/n/10 (\OML/cmm/m/it/10 L\/cmr/m/n/10 )\OML/cmm/m/it/10 u +\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )\OML/cmm/m/it/10 : + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.22 E + t, on obtient comme valeurs de $A$, $B$ et $C$ : +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no E in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 22--23 +[] + [] + + +Overfull \hbox (7.50002pt too wide) in paragraph at lines 22--23 +\OML/cmm/m/it/10 A$ + [] + + +Overfull \hbox (8.0868pt too wide) in paragraph at lines 22--23 +\OML/cmm/m/it/10 B$ + [] + + +Overfull \hbox (7.86249pt too wide) in paragraph at lines 22--23 +\OML/cmm/m/it/10 C$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.24 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 24--24 +[] + [] + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.26 \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.27 0 & + E^{-1}D \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.28 -E^{-1}D^T & + 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 24 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.29 \end{bmatrix} + = +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.30 \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.31 0 & + 0 & 0 & 0 & 114 & 6 & 12 & -2 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.31 0 & 0 & + 0 & 0 & 114 & 6 & 12 & -2 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.31 0 & 0 & 0 & + 0 & 114 & 6 & 12 & -2 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.31 0 & 0 & 0 & 0 & + 114 & 6 & 12 & -2 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.31 0 & 0 & 0 & 0 & 114 & + 6 & 12 & -2 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.31 0 & 0 & 0 & 0 & 114 & 6 & + 12 & -2 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.31 0 & 0 & 0 & 0 & 114 & 6 & 12 & + -2 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.32 0 & + 0 & 0 & 0 & -54 & -6 & -2 & 4 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.32 0 & 0 & + 0 & 0 & -54 & -6 & -2 & 4 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.32 0 & 0 & 0 & + 0 & -54 & -6 & -2 & 4 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.32 0 & 0 & 0 & 0 & + -54 & -6 & -2 & 4 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.32 0 & 0 & 0 & 0 & -54 & + -6 & -2 & 4 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.32 0 & 0 & 0 & 0 & -54 & -6 & + -2 & 4 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.32 0 & 0 & 0 & 0 & -54 & -6 & -2 & + 4 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.33 0 & + 0 & 0 & 0 & -1188 & -12 & -114 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.33 0 & 0 & + 0 & 0 & -1188 & -12 & -114 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.33 0 & 0 & 0 & + 0 & -1188 & -12 & -114 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.33 0 & 0 & 0 & 0 & + -1188 & -12 & -114 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.33 0 & 0 & 0 & 0 & -1188 & + -12 & -114 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.33 0 & 0 & 0 & 0 & -1188 & -12 & + -114 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.33 0 & 0 & 0 & 0 & -1188 & -12 & -114 & + 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.34 0 & + 0 & 0 & 0 & -828 & -12 & -54 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.34 0 & 0 & + 0 & 0 & -828 & -12 & -54 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.34 0 & 0 & 0 & + 0 & -828 & -12 & -54 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.34 0 & 0 & 0 & 0 & + -828 & -12 & -54 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.34 0 & 0 & 0 & 0 & -828 & + -12 & -54 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.34 0 & 0 & 0 & 0 & -828 & -12 & + -54 & 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.34 0 & 0 & 0 & 0 & -828 & -12 & -54 & + 6 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.35 6 & + 54 & 4 & -2 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.35 6 & 54 & + 4 & -2 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.35 6 & 54 & 4 & + -2 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.35 6 & 54 & 4 & -2 & + 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.35 6 & 54 & 4 & -2 & 0 & + 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.35 6 & 54 & 4 & -2 & 0 & 0 & + 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.35 6 & 54 & 4 & -2 & 0 & 0 & 0 & + 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.36 -6 & + -114 & -2 & 12 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.36 -6 & -114 & + -2 & 12 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.36 -6 & -114 & -2 & + 12 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.36 -6 & -114 & -2 & 12 & + 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.36 -6 & -114 & -2 & 12 & 0 & + 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.36 -6 & -114 & -2 & 12 & 0 & 0 & + 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.36 -6 & -114 & -2 & 12 & 0 & 0 & 0 & + 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.37 -12 & + -828 & -6 & 54 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.37 -12 & -828 & + -6 & 54 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.37 -12 & -828 & -6 & + 54 & 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.37 -12 & -828 & -6 & 54 & + 0 & 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.37 -12 & -828 & -6 & 54 & 0 & + 0 & 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.37 -12 & -828 & -6 & 54 & 0 & 0 & + 0 & 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.37 -12 & -828 & -6 & 54 & 0 & 0 & 0 & + 0 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.38 -12 & + -1188 & -6 & 114 & 0 & 0 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.38 -12 & -1188 & + -6 & 114 & 0 & 0 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.38 -12 & -1188 & -6 & + 114 & 0 & 0 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.38 -12 & -1188 & -6 & 114 & + 0 & 0 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.38 -12 & -1188 & -6 & 114 & 0 & + 0 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.38 -12 & -1188 & -6 & 114 & 0 & 0 & + 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.38 -12 & -1188 & -6 & 114 & 0 & 0 & 0 & + 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 24 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.39 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (728.30418pt too wide) detected at line 40 +\OML/cmm/m/it/10 A \/cmr/m/n/10 = 0\OML/cmm/m/it/10 E[]D[] \OMS/cmsy/m/n/10 ^^ +@ \OML/cmm/m/it/10 E[]D[]\/cmr/m/n/10 0 = 0000114612 \OMS/cmsy/m/n/10 ^^@ \/cmr +/m/n/10 2[] 0000 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 54 \OMS/cmsy/m/n/10 ^^@ \/cm +r/m/n/10 6 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 24[] 0000 \OMS/cmsy/m/n/10 ^^@ \/c +mr/m/n/10 1188 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 12 \OMS/cmsy/m/n/10 ^^@ \/cmr/ +m/n/10 1146[] 0000 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 828 \OMS/cmsy/m/n/10 ^^@ \ +/cmr/m/n/10 12 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 546[] 6544 \OMS/cmsy/m/n/10 ^^ +@ \/cmr/m/n/10 20000[] \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 6 \OMS/cmsy/m/n/10 ^^ +@ \/cmr/m/n/10 114 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 2120000[] \OMS/cmsy/m/n/1 +0 ^^@ \/cmr/m/n/10 12 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 828 \OMS/cmsy/m/n/10 ^^ +@ \/cmr/m/n/10 6540000[] \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 12 \OMS/cmsy/m/n/10 + ^^@ \/cmr/m/n/10 1188 \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 61140000 + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.42 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 42--42 +[] + [] + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.44 \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{equation} on input line 42 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.46 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.48 \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{equation} on input line 42 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.57 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.60 \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.61 0 & + -\Phi^{T}(L,t) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 42 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.62 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.64 \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.65 0 & + 0 & 0 & 0 & 0 & -1 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.65 0 & 0 & + 0 & 0 & 0 & -1 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.65 0 & 0 & 0 & + 0 & 0 & -1 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.65 0 & 0 & 0 & 0 & + 0 & -1 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.65 0 & 0 & 0 & 0 & 0 & + -1 & 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.65 0 & 0 & 0 & 0 & 0 & -1 & + 0 & 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.65 0 & 0 & 0 & 0 & 0 & -1 & 0 & + 0 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 42 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.66 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (302.61017pt too wide) detected at line 67 +\OML/cmm/m/it/10 B \/cmr/m/n/10 = 0[] \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 E[ +]\/cmr/m/n/10 ^^H(\OML/cmm/m/it/10 L\/cmr/m/n/10 ) = 0[] 0[] 0[] 0[] 4[] \OMS/ +cmsy/m/n/10 ^^@ \/cmr/m/n/10 16[] \OMS/cmsy/m/n/10 ^^@ \/cmr/m/n/10 60[] \OMS +/cmsy/m/n/10 ^^@ \/cmr/m/n/10 120 \OML/cmm/m/it/10 C \/cmr/m/n/10 = 0 \OMS/cmsy +/m/n/10 ^^@ \/cmr/m/n/10 ^^H[](\OML/cmm/m/it/10 L; t\/cmr/m/n/10 ) = 00000 \OMS +/cmsy/m/n/10 ^^@ \/cmr/m/n/10 100 + [] + +Missing character: There is no F in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.70 \begin{equation} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 70--70 +[] + [] + +! Undefined control sequence. +l.71 \boxed + { +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Environment aligned undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.72 \begin{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.73 \dot{x}_d(t) & + = Ax_d(t)+Bu(t),\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.74 y(t) & + = Cx_d(t). +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{equation} on input line 70 ended by \end{aligned}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.75 \end{aligned} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (168.62527pt too wide) detected at line 77 +[] + [] + +) +! Emergency stop. +<*> Q3.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 40 strings out of 474222 + 726 string characters out of 5748733 + 1925975 words of memory out of 5000000 + 22393 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,114b,91s stack positions out of 10000i,1000n,20000p,200000b,200000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/latex/Questions/Q3.tex b/latex/Questions/Q3.tex index 703e6de..8a29caa 100644 --- a/latex/Questions/Q3.tex +++ b/latex/Questions/Q3.tex @@ -24,6 +24,10 @@ Et, on obtient comme valeurs de $A$, $B$ et $C$ : \begin{equation} A = \begin{bmatrix} +0 & E^{-1}D \\ +-E^{-1}D^T & 0 +\end{bmatrix} = +\begin{bmatrix} 0 & 0 & 0 & 0 & 114 & 6 & 12 & -2 \\ 0 & 0 & 0 & 0 & -54 & -6 & -2 & 4 \\ 0 & 0 & 0 & 0 & -1188 & -12 & -114 & 6 \\ @@ -38,6 +42,10 @@ A = \begin{equation} B= \begin{bmatrix} +0 \\ -E^{-1}\Phi(L) +\end{bmatrix} += +\begin{bmatrix} 0\\ 0\\ 0\\ @@ -50,6 +58,10 @@ B= \qquad C= \begin{bmatrix} +0 & -\Phi^{T}(L,t) +\end{bmatrix} += +\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \end{bmatrix} \end{equation} diff --git a/latex/Questions/Q4.tex b/latex/Questions/Q4.tex index 630c08a..95ed348 100644 --- a/latex/Questions/Q4.tex +++ b/latex/Questions/Q4.tex @@ -1,10 +1,6 @@ - \textbf{Question 4} -La Figure 1a correspond aux valeurs propres en temps continu, tandis que les figures 1b, 1c et 1d montrent les valeurs propres discrètes obtenues avec les méthodes de Tustin et ZOH pour différents temps d'échantillonnage. - -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. Le système discret est donc à la limite de stabilité. +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 @@ -37,3 +33,8 @@ Après discrétisation, les pôles obtenus avec les méthodes de Tustin et ZOH s \caption{Comparaison des valeurs propres du système en temps continu et en temps discret.} \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. + + diff --git a/latex/Questions/Q5.tex b/latex/Questions/Q5.tex index 151f202..cc21bac 100644 --- a/latex/Questions/Q5.tex +++ b/latex/Questions/Q5.tex @@ -1,9 +1,10 @@ +\newpage \textbf{Question 5} La Figure 2 montre que la sortie $y(t)$ oscille après l’échelon, confirmant l’instabilité du système en boucle ouverte. \begin{figure}[H] \centering - \includegraphics[width=0.4\textwidth]{Illustrations/Q5stepReponse} % Remplacez par votre image + \includegraphics[scale=0.4]{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 14a4f73..de72eeb 100644 --- a/latex/Questions/Q7.tex +++ b/latex/Questions/Q7.tex @@ -1,28 +1,12 @@ \textbf{Question 7} +Nous avons implementé la loi de commande: $u(t) = -Ky(t) + Hy_{c}(t)$ Nous avons utilisé une boucle \texttt{for} sous MATLAB afin de tester les valeurs de $k$ entre 0 e 100. -Le gain $H$ est ensuite calculé à partir d'un système en boucle fermée de la forme : - -\begin{equation} -\dot{x}(t) = (A-BK)x(t) + BHr(t), -\qquad -y(t)=Cx(t), -\end{equation} - -on impose que, en régime permanent, la sortie suive la consigne. On utilise donc : - -\begin{equation} -H = -\frac{1}{C(A-BK)^{-1}B}. -\end{equation} - +Le gain $H$ est ensuite calculé à partir d'un système en boucle fermée sous la forme: +\begin{center} +$H=-\frac{1}{C(A-BKC)^{-1}B}$ +\end{center} Ainsi, les valeurs retenues pour la commande en boucle fermée sont : - -\begin{equation} -\boxed{ -k = 1{,}85, -\qquad -H = -3. -} -\end{equation} \ No newline at end of file +$K = 1{,}85$ \textit{\&} $H = -3$ diff --git a/latex/Questions/Q8.tex b/latex/Questions/Q8.tex index f9f04a5..61f3135 100644 --- a/latex/Questions/Q8.tex +++ b/latex/Questions/Q8.tex @@ -1,9 +1,10 @@ \textbf{Question 8} -La figure montre que $w(L,t)$ suit la consigne $w_c(t)$ et converge vers 1, avec une erreur nulle en régime permanent. +La Figure 3 montre que $w(L,t)$ suit la consigne $w_c(t)$ unitaire, avec une erreur nulle en régime permanent, et un temps caractéristique à $\tau = 1s$. \begin{figure}[H] \centering - \includegraphics[width=0.4\textwidth]{Illustrations/Q8} % Remplacez par votre image - \caption{Suivi de la consigne $wc(t)$ par la position $w(L,t)$.} + \includegraphics[scale=0.4]{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/contents.tex b/latex/contents.tex index 1014c5d..4ce5f09 100644 --- a/latex/contents.tex +++ b/latex/contents.tex @@ -9,6 +9,7 @@ \subsection{Discrétisation par éléments finis} \input{Questions/Q2} \input{Questions/Q3} +\newpage \input{Questions/Q4} \input{Questions/Q5} \input{Questions/Q6} @@ -23,6 +24,7 @@ %%newpage pour les imgs q9 \newpage \input{Questions/Q9} +\newpage \input{Questions/Q10} diff --git a/latex/main.aux b/latex/main.aux index 4414ae5..5fa9b14 100644 --- a/latex/main.aux +++ b/latex/main.aux @@ -2,41 +2,45 @@ \providecommand\hyper@newdestlabel[2]{} \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} -\providecommand 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