From 3e50bccb556e6a272fa54265c435ec0fad2c147c Mon Sep 17 00:00:00 2001 From: taban Date: Fri, 8 May 2026 02:45:10 +0200 Subject: [PATCH] =?UTF-8?q?Dsl=20j'avais=20oubli=C3=A9=20de=20push=20mais?= =?UTF-8?q?=20le=20projet=20est=20fini?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- latex/contents.tex | 145 +++++++++++-------------------------- latex/contents/annexes.tex | 120 ++++++++++++++++++++++++------ latex/main.pdf | Bin 573334 -> 570788 bytes 3 files changed, 142 insertions(+), 123 deletions(-) diff --git a/latex/contents.tex b/latex/contents.tex index f49e516..5860e59 100644 --- a/latex/contents.tex +++ b/latex/contents.tex @@ -60,55 +60,24 @@ Voir annexe. \subsection{Question 4} -\begin{equation*} - \omega(p, t) = C_\omega(p)x_d(t) -\end{equation*} - -\begin{equation*} - C_\omega(p) = \begin{bmatrix} - \frac{p^2(2p^3-5p^2+10)}{20} & -\frac{p^4(2p-5)}{20} & \frac{p^3(3p^2-10p+10)}{60} & \frac{p^4(3p-5)}{60} & 0 & 0 & 0 & 0 - \end{bmatrix} -\end{equation*} - -\begin{equation*} - \phi(L)^T = \begin{bmatrix} - 2p^3 - 3p^2 + 1 & 3p^2 - 2p^3 & p^3 - 2p^2 + p & p^3 - p^2 - \end{bmatrix} -\end{equation*} - +\vspace*{-1.5em} \begin{figure}[H] \centering \includegraphics[width=\textwidth]{Illustrations/Question4} \end{figure} - +\vspace*{-0.5em} Les valeurs propres du système dans tous les cas (continu, discrétisé avec Tustin et bloqueur d'ordre 0) sont dans les zones de stabilité. La stabilité du système est vérifiée pour tous les cas. \subsection{Question 5} +\vspace*{-2em} \begin{figure}[H] \centering \includegraphics[width=35em]{Illustrations/Question5} \end{figure} -\begin{align*} - y = -\phi(L)^Te_{2d} && y = -e_2(L, t) & & e_2(\zeta, t) \approx \phi(\zeta)^Te_{2d}(t) -\end{align*} - -\begin{align*} - x_1 = \frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t)\approx\phi(\zeta)^2x_{1d} & & x_2(\zeta,t) = \rho(\zeta)\frac{\partial\omega}{\partial t}(\zeta,t) \approx\phi(\zeta)^Tx_{2d} -\end{align*} - -\begin{align*} - \begin{bmatrix} - \dot{x}_{1d} \\ \dot{x}_{1d} - \end{bmatrix} = A \begin{bmatrix} - x_{1d} \\ x_{2d} - \end{bmatrix} + Bu & & y = C \begin{bmatrix} - x_{1d} \\ x_{2d} - \end{bmatrix} -\end{align*} - \subsection{Question 6} +\vspace*{-2em} \begin{equation*} \begin{bmatrix} x_1 \\ x_2 @@ -122,13 +91,13 @@ Les valeurs propres du système dans tous les cas (continu, discrétisé avec Tu x_{1d} \\ x_{2d} \end{bmatrix} \end{equation*} - +\vspace*{-.4em} \begin{equation*} \frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t) = \underbrace{\begin{bmatrix} \phi(\zeta)^T & 0 & 0 & 0 & 0 \end{bmatrix}}_{C.I.=0+0}x_{d}(t) \end{equation*} - +\vspace*{-1em} \begin{equation*} \begin{aligned} \frac{\partial\omega}{\partial\zeta}(\zeta,t) &= \int\frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t)d\zeta = x_d(t)\int\begin{bmatrix} @@ -140,76 +109,54 @@ Les valeurs propres du système dans tous les cas (continu, discrétisé avec Tu \end{aligned} \end{equation*} Du côté gauche de la poutre soit $\zeta = 0$, la vitesse et la déformation de la poutre sont nulles. Grâce à ces deux données, nous pouvons calculer les constantes représentant les conditions -initiales qui sont donc aussi nulles. -En calculant les intégrales de $\phi(\zeta)$, nous retrouvons les coefficients de $C_w(\zeta)$. Nous avons alors bien $w(\zeta ,t) = C_w(\zeta) x_d (t)$. +initiales qui sont donc aussi nulles. En calculant les intégrales de $\phi(\zeta)$, nous retrouvons les coefficients de $C_w(\zeta)$. Nous avons alors bien $w(\zeta ,t) = C_w(\zeta) x_d (t)$. + - \section{Retour de sortie} - +\vspace*{-.5em} \subsection{Question 7} -La sortie est donnée par $y(t) = C\,x_d(t)$, donc le système en boucle fermée s’écrit : - -\begin{equation*} -\begin{cases} - \dot{x}_d(t) = (A - B C k)\,x_d(t) + B H\,w_c(L,t) \\ - w(L,t) = C_w(L)\,x_d(t) -\end{cases} -\end{equation*} - -En régime permanent, $\dot{x}_d(t)=0$, donc : -\begin{equation*} - 0 = (A - B C k)\,x_d + B H\,C_w(L)\,x_d - \quad \Rightarrow \quad - x_d = -(A - B C k)^{-1} B H w_c(L,t) -\end{equation*} - -En multipliant l’expression par $C_w(L)$, on trouve : -\begin{equation*} - w(L,t) = C_w(L)\,x_d = -C_w(L)(A - B C k)^{-1} B H w_c(L,t) -\end{equation*} - -Pour assurer le suivi $w(L,t) = w_c(L,t)$, il faut : -\begin{equation*} - - C_w(L)(A - B C k)^{-1} B H = I -\end{equation*} - -D’où : +\vspace*{-.2em} +Suivant les calculs de l'annexe, nous avons : \begin{equation*} \boxed{H = -\left(C_w(L)(A - B C k)^{-1} B\right)^{-1}} \end{equation*} -Finalement, après calcul sous MATLAB, on obtient $H = -3$. +Après calcul dans MATLAB, nous obtienons $\boxed{H = -3}$. \subsection{Question 8} +\vspace*{-2em} \begin{figure}[H] \centering - \includegraphics[width=35em]{Illustrations/Question8} + \includegraphics[width=30em]{Illustrations/Question8} \end{figure} +\vspace*{-1em} \subsection{Question 9} +\vspace*{-2em} \begin{figure}[H] \centering - \includegraphics[width=35em]{Illustrations/Question9} + \includegraphics[width=25em]{Illustrations/Question9} \end{figure} +\vspace*{-1em} En discrétisant notre système par la période d'échantillonnage $T_{s3} = 0.04 \ [s]$, on obtient une valeur propre en dehors du cercle unitaire de stabilité. Cette période d'échantillonnage déstabilise le système. \subsection{Question 10} +\vspace*{-2em} +\begin{figure}[H] + \centering + \includegraphics[width=30em]{Illustrations/Question10} +\end{figure} +\vspace*{-1em} La période d'échantillonnage maximale permettant de garantir la stabilité asymptotique est de $T_s \approx 0.011 \ [s]$. -\begin{figure}[H] - \centering - \includegraphics[width=35em]{Illustrations/Question10} -\end{figure} - - \section{Retour d'état} - +\vspace*{-1em} \subsection{Question 11} -En utilisant la fonction \texttt{lqr()} de MATLAB avec $Q = I_8$ et $R = 1$, on trouve la matrice de gain $K$ suivante : +En utilisant la fonction \texttt{lqr()} de MATLAB avec $Q = I_8$ et $R = 1$, on trouve la matrice de gain : \begin{equation*} K = \begin{pmatrix} 0.97 & -14.62 & 0.66 & 1.32 & 19.32 & 0.39 & 2.40 & -2.11 @@ -217,6 +164,7 @@ En utilisant la fonction \texttt{lqr()} de MATLAB avec $Q = I_8$ et $R = 1$, on \end{equation*} \subsection{Question 12} +\vspace*{-2em} \begin{figure}[H] \centering \includegraphics[width=35em]{Illustrations/Question12} @@ -227,7 +175,7 @@ En utilisant la fonction \texttt{lqr()} de MATLAB avec $Q = I_8$ et $R = 1$, on La même période d'échantillonnage que la section précédente n'est pas appropriée pour cette loi de commande. La période d'échantillonnage maximale permettant de garantir la stabilité asymptotique est de $T_s = 0.0107 \ [s]$. - +\vspace*{-1em} \begin{figure}[H] \centering \includegraphics[width=35em]{Illustrations/Question13} @@ -237,34 +185,23 @@ est de $T_s = 0.0107 \ [s]$. \subsection{Question 14} En prenant une période d'échantillonnage $T_s = 0.01 \ [s]$, on se rapproche du comportement désiré de la question 12. +\vspace*{-1em} \begin{figure}[H] \centering \includegraphics[width=35em]{Illustrations/Question14} \end{figure} - +\vspace*{-1em} \section{Rejection de perturbation} - +\vspace*{-1em} \subsection{Question 15} -On sait que : +\vspace*{-2em} \begin{equation*} - F_{ext} = \int_{0}^{L}\phi(\zeta)q(\zeta,t)d\zeta - \quad \Rightarrow \quad - F_{ext} = q_0(t) \cdot \int_{0}^{L}\phi(\zeta)d\zeta \text{ car } q(\zeta,t) = q_0(t) -\end{equation*} -Or on a : -\begin{equation*} - E\dot{x}_{2d} = - e_{1d}D^T - \phi(L)u(t) - F_{ext} -\end{equation*} -Donc, puisque $\dot{x}_{d}(t) = A x_d(t) + B u(t) + B_p q_0(t)$, on trouve : -\begin{equation*} -B_p = -\begin{pmatrix} +B_p = \begin{pmatrix} 0_{4 \times 1} \\ -\int_{0}^{L}\phi(\zeta)d\zeta \end{pmatrix} -= -\begin{pmatrix} += \begin{pmatrix} 0_{4 \times 1} \\ \frac{1}{2} \\ \frac{1}{2} \\ @@ -272,12 +209,14 @@ B_p = -\frac{1}{12} \end{pmatrix} \end{equation*} - +\vspace*{-1em} \subsection{Question 16} +\vspace*{-2em} \begin{figure}[H] \centering \includegraphics[width=35em]{Illustrations/Question16} \end{figure} +\vspace*{-1em} Notre système ne permet pas de garantir une erreur nulle en régime permanent face à une perturbation constante. @@ -286,6 +225,7 @@ Nous souhaitons garder les mêmes valeurs propres que celles obtenues lors de la avec la LQR. La valeur propre supplémentaire doit être plus à droite pour permettre de rejeter la perturbation. On choisit $\lambda_9 = -2$. On trouve alors : +\vspace*{-.5em} \begin{equation*} \lambda = \begin{pmatrix} -84.66 & -7.56 \pm 68.77i & -27.97 \pm 17.24i & -15.91 & -4.38 \pm 2.66i & -2 @@ -294,16 +234,16 @@ On trouve alors : \subsection{Question 18} En utilisant la fonction \texttt{place()} de MATLAB, on trouve la matrice de gain $K$ suivante : +\vspace*{-.5em} \begin{equation*} -K_{aug} = -\begin{pmatrix} +K_{aug} = \begin{pmatrix} K_1 & K_i \end{pmatrix} \quad \text{avec} \quad K_1 \in \mathbb{R}^{1 \times 8}, \; K_i \in \mathbb{R} \end{equation*} -\vspace*{-2em} +\vspace*{-2.8em} \begin{align*} K_1 = \begin{pmatrix} -0.89 & -16.59 & 0.28 & 1.68 & 19.39 & -0.84 & 2.36 & -1.94 @@ -315,11 +255,12 @@ K_i \in \mathbb{R} \subsection{Question 19} +\vspace*{-2em} \begin{figure}[H] \centering \includegraphics[width=35em]{Illustrations/Question19} \end{figure} - +\vspace*{-1em} En utilisant l'action intégrale, on parvient à rejeter la perturbation constante et à garantir une erreur nulle en régime permanent. diff --git a/latex/contents/annexes.tex b/latex/contents/annexes.tex index df08436..3b98b51 100644 --- a/latex/contents/annexes.tex +++ b/latex/contents/annexes.tex @@ -13,7 +13,7 @@ \newpage \pagenumbering{Alph} \section{Annexe} - +\vspace*{-1em} \subsection{Question 2} \vspace*{-2em} \begin{align*} @@ -27,8 +27,8 @@ \end{aligned}\right. \end{align*} -En utilisant la première ligne de l'équation 1, on trouve : - +En utilisant la 1\textsuperscript{ère} ligne de l'éq. 1, on trouve : +\vspace*{-.2em} \begin{equation*} \int\phi(\zeta)d\zeta\times\phi^T\dot{x_{1d}} = \int\phi(\zeta)d\zeta\times\frac{\partial^2}{\partial\zeta^2} = \ddot{\phi}(\zeta)^Te_{2d} \end{equation*} @@ -38,7 +38,7 @@ En utilisant la première ligne de l'équation 1, on trouve : \end{equation} On applique plusieurs fois de l’intégration par partie (IPP) : - +\vspace*{-.5em} \begin{equation*} \begin{aligned} \int_{0}^{L}\phi(\zeta)\ddot{\phi}(\zeta)^T d\zeta &= [\phi(\zeta)\dot{\phi}(\zeta)^T]_{0}^{L} - \int_{0}^{L}\dot{\phi}(\zeta)\dot{\phi}(\zeta)^T d\zeta \\ @@ -46,11 +46,8 @@ On applique plusieurs fois de l’intégration par partie (IPP) : &= \phi(L)\dot{\phi}(L)^T - \dot{\phi}(L)\phi(L)^T + \int_{0}^{L} \ddot{\phi}(\zeta)\phi(\zeta)^T d\zeta = D \end{aligned} \end{equation*} - -On continue avec ça en remplaçant dans l'équation 2 : $\Rightarrow \boxed{E\dot{x}_{1d} = De_{2d}}$ \\ - - -Puis on a dans la deuxième ligne de l'équation 1 : +\vspace*{-.5em} +Continuons en remplaçant dans l'équation 2 : $\Rightarrow \boxed{E\dot{x}_{1d} = De_{2d}}$ Dans la 2\textsuperscript{de} ligne de l'éq. 1 on a : \begin{equation*} \begin{aligned} @@ -58,9 +55,10 @@ Puis on a dans la deuxième ligne de l'équation 1 : \int_{0}^{L} \phi^T(\zeta)d\zeta\times\dot{x}_{2d} &= -e_{1d}\int_{0}^{L}\phi(\zeta)\ddot{\phi}(\zeta)^T d\zeta - \underbrace{\int_{0}^{L}\phi(\zeta)q(\zeta,t)d\zeta}_{=F_{ext}} \end{aligned} \end{equation*} +\vspace*{-1.3em} Puis on trouve D : - +\vspace*{-.5em} \begin{equation*} \begin{aligned} & D = \phi(L)\dot{\phi}(L)^T - \phi(L)\dot{\phi}(L)^T + \int_{0}^{L}\ddot{\phi}(\zeta)\phi(\zeta)^Td\zeta \\ @@ -69,9 +67,7 @@ Puis on trouve D : \Rightarrow & E\dot{x}_{2d} = -e_{1d}D^T + \overbrace{e_{1d}\dot{\phi}(L)\phi(L)^T}^{=0} - e_{1d}\phi(L)\dot{\phi}(L)^T - F_{ext} \end{aligned} \end{equation*} - -Or on sait que : - +\vspace*{-.8em} \begin{align*} \Rightarrow\left\{\begin{aligned} \frac{de_1}{d\zeta} = u(t) &\Rightarrow \dot{\phi}(L)^T e_{1d} = u(t) \\ @@ -82,7 +78,7 @@ Or on sait que : \end{align*} Puis on a ceci : - +\vspace*{-.1em} \begin{align*} \left\{\begin{aligned} y(t) &= -e_2 (L,t) \\ @@ -94,8 +90,7 @@ Puis on a ceci : \subsection{Question 3} -On a : - +\vspace*{-2em} \begin{align*} \left\{\begin{aligned} e_1 = EIx_1 = x_1 &\Rightarrow e_{1d} = x_{1d} \\ @@ -108,11 +103,9 @@ On a : y &= -\phi(L)^T x_{2d} \end{aligned}\right. \end{align*} - -Ce qui nous donne : - +\vspace*{-.5em} \begin{equation*} - A = \begin{pmatrix} + \Rightarrow A = \begin{pmatrix} 0 & E^{-1}D \\ -E^{-1}D^T & 0 \end{pmatrix}, \ B = \begin{pmatrix} @@ -130,8 +123,93 @@ Ce qui nous donne : \end{pmatrix} \end{aligned}\right. \end{equation*} +\vspace*{-1.5em}\\ +Puis application numérique +\vspace*{-2em}\\ +\subsection{Question 4} +\vspace*{-2em} +\begin{equation*} + \omega(p, t) = C_\omega(p)x_d(t) = \begin{bmatrix} + \frac{p^2(2p^3-5p^2+10)}{20} & -\frac{p^4(2p-5)}{20} & \frac{p^3(3p^2-10p+10)}{60} & \frac{p^4(3p-5)}{60} & 0 & 0 & 0 & 0 + \end{bmatrix}x_d(t) +\end{equation*} + +\begin{equation*} + \phi(L)^T = \begin{bmatrix} + 2p^3 - 3p^2 + 1 & 3p^2 - 2p^3 & p^3 - 2p^2 + p & p^3 - p^2 + \end{bmatrix} +\end{equation*} +\vspace*{-4em}\\ +\subsection{Question 5} +\vspace*{-2em} +\begin{align*} + y = -\phi(L)^Te_{2d} && y = -e_2(L, t) & & e_2(\zeta, t) \approx \phi(\zeta)^Te_{2d}(t) +\end{align*} +\vspace*{-2em} +\begin{align*} + x_1 = \frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t)\approx\phi(\zeta)^2x_{1d} & & x_2(\zeta,t) = \rho(\zeta)\frac{\partial\omega}{\partial t}(\zeta,t) \approx\phi(\zeta)^Tx_{2d} +\end{align*} +\vspace*{-2em} +\begin{align*} + \begin{bmatrix} + \dot{x}_{1d} \\ \dot{x}_{1d} + \end{bmatrix} = A \begin{bmatrix} + x_{1d} \\ x_{2d} + \end{bmatrix} + Bu & & y = C \begin{bmatrix} + x_{1d} \\ x_{2d} + \end{bmatrix} +\end{align*} +\vspace*{-4em}\\ + +\subsection{Question 7} + +\vspace*{-.5em} +La sortie est donnée par $y(t) = C\,x_d(t)$, donc le système en boucle fermée s’écrit : + +\begin{equation*} + \begin{cases} + \dot{x}_d(t) = (A - B C k)\,x_d(t) + B H\,w_c(L,t) \\ + w(L,t) = C_w(L)\,x_d(t) + \end{cases} +\end{equation*} + +En régime permanent, $\dot{x}_d(t)=0$, donc : +\begin{equation*} + 0 = (A - B C k)\,x_d + B H\,C_w(L)\,x_d + \quad \Rightarrow \quad + x_d = -(A - B C k)^{-1} B H w_c(L,t) +\end{equation*} + +En multipliant l’expression par $C_w(L)$, on trouve : +\begin{equation*} + w(L,t) = C_w(L)\,x_d = -C_w(L)(A - B C k)^{-1} B H w_c(L,t) +\end{equation*} + +Pour assurer le suivi $w(L,t) = w_c(L,t)$, il faut : +\begin{equation*} + - C_w(L)(A - B C k)^{-1} B H = I +\end{equation*} +\vspace*{-4em}\\ + +\subsection{Question 15} +\vspace*{-.5em} +On sait que : +\begin{equation*} + F_{ext} = \int_{0}^{L}\phi(\zeta)q(\zeta,t)d\zeta + \quad \Rightarrow \quad + F_{ext} = q_0(t) \cdot \int_{0}^{L}\phi(\zeta)d\zeta\ \text{ car }\ q(\zeta,t) = q_0(t) +\end{equation*} +Or on a $E\dot{x}_{2d} = - e_{1d}D^T - \phi(L)u(t) - F_{ext}$\\ +Donc, puisque $\dot{x}_{d}(t) = A x_d(t) + B u(t) + B_p q_0(t)$, on trouve les résultats d'en-haut. + + + + + + + + -Puis application numérique. diff --git a/latex/main.pdf b/latex/main.pdf index 3574091e2b86a6867181c0e76993494707588c7b..739974433e0d298e2bb47394f0ba8b8cfd585501 100644 GIT binary patch delta 52682 zcmZs?Wk3|)|NiYRrF1PwNq6o7OE-dacXyYhh%D05GQiRXAV^3mAPs_qlr%^vAPUk_ zBKRLZy5HaZ;C=$sR71f<;7meH`uKt&VoY!m#J{#U*cSfR zmH^u#|JraNP>5m#927{5K!8FFBZNUAju9fDkiZC0Cb*~&Mj)9`a8bU4)AZ zF`@o?584P9#~^?8y#W4t=i*Rua0_Y-K9D@M5 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