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AA +% Relu jusqu'au 2.3. 28.10.14. AA + +\documentclass{book} +\input{../../preambule/preambule} +\renewcommand{\nomentete}{UE 421 : Contrôle de processus} +\renewcommand{\thesection}{\arabic{section}} +\title{\nomentete} + + +\begin{document} + +\maketitle +\tableofcontents + + +\chapter{Echantillonnage des signaux et transformée en z} + +\section{Introduction : positionnement du problème} + +On va s'intéresser aux signaux analogiques vus comme des fonctions réelles + +\[x : t \in \R \rightarrow x(t) \in \R\] + +Les processus évoluent continûment dans le temps.\\ + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\sbEntree{E} +\sbComp{comp}{E} +\sbRelier[$c(t)$]{E}{comp} +\sbBloc{reg}{Structure de commande}{comp} +\sbRelier[$\epsilon$]{comp}{reg} +\sbBloc{sys}{Système}{reg} +\sbRelier[u(t)]{reg}{sys} +\sbSortie{S}{sys} +\sbRelier[$y(t)$]{sys}{S} +\sbRenvoi{sys-S}{comp}{} +\end{tikzpicture} +\caption{Asservissement analogique} +\end{figure} + + +La loi de commande est alors : +\begin{align*} +U(p) & = C(p).[R(p) - Y(p)], \text{ transformée de Laplace de }\\ +u(t) & = c(t)*[r(t) - y(t)] +\end{align*} + +\paragraph*{Problématique} +Il faut alors évaluer u(t) et le mettre en œuvre de manière analogique et/ou bon moment. Pour cela, il est nécessaire de calculer en temps réels et d'adapter u(t).\\ + +La solution est d'exploiter un calculateur numérique couplé à de l'électronique numérique pour implémenter la loi de commande. +Par exemple : +\begin{itemize} +\item ordinateur à base de microprocesseurs cadencés par une horloge interne +\item micro-contrôleurs +\item DSP (Digital Signal Processing, puce à usage spécifique, non modifiable) +\item Arduino +\end{itemize} + +L'information est transmise par des signaux binaires eux-mêmes étant des signaux numériques. Cette information ne transporte pas l'énergie nécessaire pour contrôler le processus, mais seulement la loi de commande. + +Les signaux numériques évoluent de manière discrète à des instants régulièrement espacés par un intervalle de temps donné par la période de l'horloge $T_h = \frac{1}{f_h}$. + +On pose l'hypothèse que $T_h$ est constante donc les différents instants correspondent à $k.T_h$ où $k\in\mathbb{N}$.\\ + +\subsection*{Définition} +Le signal numérique $u_k$ est défini comme une suite numérique : +\begin{align*} +\mathbb{N} & \rightarrow \mathbb{R} \\ +k & \mapsto U_k +\end{align*} + +\subsection*{Calculateurs numériques} +Ils servent à implémenter les lois de commande, c'est-à-dire les règles mathématiques d'évolution des signaux. + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\sbEntree{E} + +\sbBloc{calc}{Calculateur numérique}{E} +\sbRelier[$e_k$]{E}{calc} + +\sbBloc{cna}{CNA}{calc} +\sbRelier[$u_k$]{calc}{cna} + +\sbBloc{sys}{Système analogique}{cna} +\sbRelier[$u(t)$]{cna}{sys} + +\sbSortie{S}{sys} +\sbRelier[$y(t)$]{sys}{S} + +\sbDecaleNoeudy[4]{S}{R} +\sbBlocr[12]{can}{CAN}{R} +\sbRelieryx{sys-S}{can} +\sbRelierxy{can}{calc} + +\end{tikzpicture} +\caption{Interfaçage Numérique / Analogique} +\end{figure} + +\noindent Remarque :\\ +CAN : Convertisseur Analogique Numérique\\CNA : Convertisseur Numérique Analogique\\ + +L'horloge permet le fonctionnement synchrone des différents composants de la structure de l'asservissement numérique. + +\section{Modélisation des signaux échantillonnés} +\subsection*{Échantillonnage} +\begin{dfn} +Un échantillonnage idéal à la période d'échantillonnage $T_e$ est représenté par : + +\begin{figure}[h!] +\centering +\begin{circuitikz} +\draw (0,0) node[above]{$u(t)$} to [cspst,l=$T_e$](2,0) node[above]{$u^*(t)$}; +\end{circuitikz} +\caption{Échantillonnage idéal} +\end{figure} + +\[ +u^*(t) = +\left\{ +\begin{array}{ll} +u(k.T_e)=u_k & \si t =k.T_e \\ +0 & \si t\neq k.T_e +\end{array} +\right. +\] +\end{dfn} + +\subsection*{Peigne de Dirac} +\begin{dfn} +On définit le peigne de Dirac par : \[p(t)=\sum_{k\in\mathbb{N}}\delta_0(t-k.T_e)\] +\end{dfn} + +On peut donc réécrire l'expression de l'échantillonnage : + +\begin{figure}[h!] +\centering +\begin{tikzpicture}[scale=0.8,samples=50,domain=-4.5:4.5] + \draw[-stealth] (-5,0) -- (5,0) node[right] {$t$}; + \draw[-stealth] (0,-1.5) -- (0,1.5) node[above] {$p(t)$}; + \foreach \n in {-4,-3,...,4} + \draw[-stealth,thick] (\n,0) -- (\n,1); + \draw (1,0) node[below]{$T_e$}; +\end{tikzpicture} +\begin{tikzpicture}[scale=0.8,samples=50,domain=-4.5:4.5] + \draw[-stealth] (-5,0) -- (5,0) node[right] {$t$}; + \draw[-stealth] (0,-1.5) -- (0,1.5) node[above] {$p^*(t)$}; + \draw[dotted] plot (\x,{sin(60*\x+20)}); + \foreach \n [evaluate=\n as \x using sin(60*\n+20)] in {-4,-3,...,4} + \draw[-stealth,thick] (\n,0) -- (\n,\x); + +\end{tikzpicture} +\caption{Peigne de Dirac et échantillonnage d'un signal} +\end{figure} + +\begin{align*} +u^*(t) &= u(t).p(t)\\ +&=\sum_{k\in\mathbb{N}}u(t)\delta_0(t-k.T_e) \\ +&=\sum_{k\in\mathbb{N}}u(kT_e)\delta_0(t-k.T_e) \\ +u^*(t) &= \sum_{k\in\mathbb{N}} u_k\delta_0(t-k.T_e) \\ +\end{align*} + +\section{Transformée en $z$ et lien avec Fourier / Laplace} + +Soit $f(t)$ un signal. +\begin{align*} +\text{Transformée de Laplace : } & L\{f(t)\} = \int_0^{\infty}f(t)e^{-pt}dt \\ +\text{Signal échantilloné : } & f^*(t) = \sum_{k\in\mathbb{N}}f_k\delta_ 0(t-kT_e) \\ +\end{align*} + +On calcule la transformée de Laplace du signal échantillonné : +\begin{align*} +F^*(p) & = L\{f^*(t)\} \\ +& = \int_ 0^{\infty}\sum_{k\in\mathbb{N}}f_k\delta_ 0(t-kT_e)e^{-tp}dt \\ +& = \int_ 0^{\infty}\sum_{k\in\mathbb{N}}f_ke^{-kT_ep}\delta_ 0(t-kT_e)dt \\ +& = \sum_{k\in\mathbb{N}}f_ke^{-kT_ep} \\ +F^*(p) & = \sum_{k\in\mathbb{N}}f_k(e^{-T_ep})^{-k} \\ +\end{align*} + +En notant $z = e^{T_ep}$, on obtient +\[ \boxed{ F(z) = F^*(p)|_{z=e^{T_ep}} } \] + +\subsection*{Transformée en $z$} +\begin{dfn} +On définit la transformée en $z$ du signal numérique $f_k$ : +\[ F(z) = \sum_{k = 0}^{\infty} f_kz^{-k} \quad, z = e^{T_ep}\] + +On note $F(z) = Z\{f_k\}$ +\end{dfn} + +\begin{prop} +La transformée en z est \textbf{linéaire} : \[ Z\{\alpha u_k + \beta f_k\} = \alpha U(z) + \beta F(z) \] + +Si $R_u$ et $R_f$ sont les rayons de convergence de $U(z)$ et de $F(z)$, alors \[R_{\alpha_u + \beta f} = \max \{ R_u,R_f \} \] +\end{prop} + +\subsection*{Produit de convolution} +\begin{dfn} +On définit le produit de convolution entre deux signaux $u_k$ et $f_k$ : +\begin{align*} +u_k * f_k & = \sum_{n=-\infty}^{\infty} u_n f_{k-n} \\ +& = \sum_{n=0}^{\infty} u_n f_{k-n} \text{ pour u et f causaux } \\ +Z\{u_k * f_k\} & = U(z).F(z) +\end{align*} +\end{dfn} + +\subsection*{Théorèmes importants} +\begin{multicols}{2} +\begin{thm}[Théorème d'avance] +\[ Z\{u_{k+d|d\in\mathbb{N}^*}\} = z^d U(z) - z^d \sum_{i=0}^{d-1}u_iz^{-i} \] +\end{thm} + +\begin{thm}[Théorème du retard] +\[ Z\{u_{k-d|d\in\mathbb{N}^*}\} = z^{-d} U(z) \] +\end{thm} +\end{multicols} + +\begin{thm}[Théorème de la sommation] +\[ Z\{\sum_{k=0}^nu_k\} = \frac{z}{z-1}U(z) \] +\end{thm} + +\begin{thm}[Théorème de la valeur initiale] +\[ \lim_{k\rightarrow 0} u_k = \lim_{z\rightarrow \infty} U(z) \] +\end{thm} + +\begin{thm}[Théorème de la valeur finale] +\[ \lim_{k\rightarrow \infty} u_k = \lim_{z\rightarrow 1} \frac{z-1}{z} U(z) \] +Cette limite est définie lorsque les pôles de $\frac{z-1}{z}U(z)$ sont à l'intérieur du cercle de rayon 1. +\end{thm} + + +\begin{prop}[Multiplication par le temps] +Soit $x(t)=te(t)$.\\ +\begin{align*} +x^*(nT_e) = x_n = nT_e e_n \\ +X(z) = Z[x_n] = -z T_e \frac{\partial E(z)}{\partial z} +\end{align*} +\end{prop} + +\subsection*{Lien avec la transformée de Fourier} +On peut considérer le peigne de Dirac $p(t)$ comme une fonction $T_e$-périodique, donc on peut la décomposer en série de Fourier : +\begin{align*} +p(t) & = \sum_{n=0}^{\infty} \delta_0(t-nT_e) = \sum_{k=-\infty}^{\infty} c_k e^{j\frac{2\pi k t}{T_e}} \\ +\text{où } c_k & = \int_{-T_e/2}^{T_e/2} (\sum_{n=0}^{\infty} \delta_0(t-nT_e)e^{-j \frac{2\pi kt}{T_e}})dt = ... = \frac{1}{T_e} \\ +\intertext{Ainsi,} + f^*(t) & = f(t).p(t) = \frac{1}{T_e} \sum_{k=-\infty}^{\infty}f(t)e^{-j\frac{2\pi kt}{T_e}} \\ +F^*(p) & = \frac{1}{T_e} \int_0^{\infty} ( \sum_{k=-\infty}^{\infty}f(t)e^{-j\frac{2\pi kt}{T_e}} ) e^{-pt}dt \\ +& = \frac{1}{T_e} \sum_{k=-\infty}^{\infty} (\int_0^{\infty} f(t) e^{-(p-j\frac{2\pi k}{T_e})t}dt) \\ +F^*(p) &= \frac{1}{T_e} \sum_{k=- \infty}^{\infty}F(p-j\frac{2\pi k}{T_e}) +\end{align*} +Le spectre de $f^*(t)$ est périodique en fréquence, de période $\frac{2\pi}{T_e}$. + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\begin{axis}[ + xlabel=$\omega$, + xmin=-2, + xmax=2, + xtick={-2,-1,0,1,2}, + xticklabels={$-4\frac{\pi}{T_e}$,$-2\frac{\pi}{T_e}$,$0$,$2\frac{\pi}{T_e}$,$4\frac{\pi}{T_e}$}, + ylabel=$|F(j\omega)|$, + ytick={0}, + axis x line*=middle, + axis y line*=middle + ] + \addplot[red,mark=none,smooth,samples=50] {gauss(0,0.5)}; + \addplot[blue,mark=none,smooth,samples=50] {gauss(1,0.5)}; + \addplot[blue,mark=none,smooth,samples=50] {gauss(-1,0.5)}; +\end{axis} +\end{tikzpicture} +\caption{Exemple d'un spectre périodique} +\end{figure} + +\subsection*{Reconstitution d'un signal} +\begin{thm} +Un signal analogique $f(t)$ dont la transformée de Fourier est nulle à l'extérieur de l'intervalle $[-\omega_0,\omega_0]$, $\omega_0>0$, est parfaitement défini par ses échantillons $f_k=f(kT_e)$ si +\[F_e=\frac{1}{T_e} \text{ vérifie } \omega_e > 2 \omega_0 \text{ : Condition de Shannon } \] + +Dans ce cas, on peut reconstituer le signal : +\[ f(t)=\sum_{k=-\infty}^{\infty}f_k sinc(\omega_e\frac{t-kT_e}{2}) \] +\end{thm} + +Preuve : à base de développement en série de Fourier.\\ + +\noindent Remarque : la méthode de reconstruction de f(t) n'est pas causale car elle suppose de connaître le signal à tout instant. +En pratique, on préfère utiliser un CNA pour des applications en temps réel. + +\section{Transformée en z inverse} + +Soit $F(z) : \mathbb{C} \rightarrow \mathbb{C}$ une fraction rationnelle propre (degré du numérateur $<$ degré du dénominateur). + +\subsection*{Problématique} Déterminer les échantillons $(f_k)_{k\in\mathbb{N}}$ tel que \[F(z) = Z\{(f_k)_{k\in\mathbb{N}}\}= \sum_{k=0}^{\infty}f_kz^{-k}\] + +\subsection{Méthode par décomposition en éléments simples} + +On applique cette méthode à $\frac{F(z)}{z}$ plutôt qu'à $F(z)$, en utilisant les transformées usuelles.\\ + +\noindent \textbf{Cas où F(z) possède des pôles distincts non nuls} + +\begin{align*} +F(z) & = \frac{...}{(z-p_1)...(z-p_ n)} \\ +\frac{F(z)}{z} & = \frac{...}{z(z-p_1)...(z-p_n)} \\ +& = \frac{C_0}{z} + \frac{C_1}{z-p_1} + ... \\ +F(z) & = C_0 + \frac{C_1 z}{z-p_1} + ... \text{ où } C_0 = F(0) \text{ et } C_i = \lim_{z\rightarrow p_i} \frac{z-p_i}{z}F(z) +\end{align*} + +De plus, on a \[ \boxed{Z^{-1}\{\frac{C_jz}{z-p_j}\}=C_jp_j^k \text{ pour } i \leq j \leq n } \] + +Exemple : +\[W(z) = \frac{z+3}{(z-1)(z+2)} \] + +\begin{align*} +\intertext{Décomposition en éléments simples} +\frac{W(z)}{z} & = \frac{z+3}{z(z-1)(z+2)} = -\frac{3/2}{z}+\frac{4/3}{z-1}+\frac{1/6}{z+2} \\ +W(z) & = -\frac{3}{2} + \frac{4}{3}\frac{z}{z-1} + \frac{1}{6}\frac{z}{z+2} \\ +w_k & = -\frac{3}{2}\delta_k + (\frac{4}{3} + \frac{1}{6}(-2)^k).\mathbf{1}_k +\end{align*} + +\noindent \textbf{Cas où F(z) possède un pôle multiple non nul, de multiplicité l supérieure ou égale à 1} + +\[ W(z) = ... + \frac{C_1z}{z-p} + \frac{C_2z}{(z-p)^2} + ... + \frac{C_lz}{(z-p)^l} + ... \] + +On a alors, $\forall j = 0,1,...,l-1$ +\[ C_{l-j} = \lim_{z \rightarrow p} (\frac{1}{j!} \frac{\partial ^j \frac{(z-p)^l}{z}W(z)}{\partial z^j}) \] + +\[Z^{-1}\{\frac{z}{(z-p)^l}\} = \frac{1}{(l-1)!}k(k-1)...(k-l+2)p^{k-l+1}, k\geq 0 \] + +Exemple dans le poly.\\ + +\noindent \textbf{Cas d'un pôle nul, de multiplicité l supérieure ou égale à 1} + +\[ W(z) = ... + C_0 + \frac{C_1z}{z} + \frac{C_2z}{z^2} + ... \] +On a alors, $\forall j = 0,1,...,l$ +\[ C_{l-j} = \lim_{z \rightarrow p} (\frac{1}{j!} \frac{\partial ^j \frac{z^l}{z}W(z)}{\partial z^j}) \] +\[ Z^{-1} [z^{-d}] = \delta_{k-d} \] + +Remarque : +\[ \frac{z}{z-D} = \frac{1}{1-Dz^{-1}} = \lim_{N\rightarrow\infty} \sum_{k=0}^N (z^{-1}D)^k = \sum_{k=0}^{\infty} D^k z^{-k} \] +On en déduit facilement que $Z^{-1}\{\frac{z}{z-D}\} = D^k$ + +\subsection{Méthode des résidus} +Méthode non exigée, voir polycopié + +\section{Modélisation des CAN et CNA} + +\subsection{Convertisseur analogique numérique} + +Problématique : +\[ y(t) \rightarrow \boxed{\text{ CAN }} \rightarrow y_k \] + +\begin{enumerate} +\item Échantillonnage : (discrétisation de l'axe des abscisses) + +on échantillonne sur les instants $kT_e$ + +\item Quantification du signal : (discrétisation de l'axe des ordonnées) + +On a $ q = \frac{u_{MAX} - u_{MIN}}{2^n}$, avec n le nombre de bits de codages, indiquant le qualité, la précision du convertisseur. +\end{enumerate} + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.6]{CAN.png} +\caption{Discrétisation et échantillonnage} +\end{figure} + +La limitation d'amplitude est source de saturation du signal échantillonné. La quantification génère un bruit sur le signal en sortie du CAN (appelé bruit de quantification). Ce bruit peut être modélisé par une variable aléatoire de moyenne nulle, de répartition uniforme et de variance donnée par $q^2 / 12 $.\\ + +Dans le cadre de ce cours, on fera l'hypothèse que la quantification ne génère pas de bruit de quantification. Il n'y auras pas non plus de saturation : on parle de numérisation parfaite.\\ + +Remarque : ces opérations induisent également des retards de l'information. \\ + +Conséquence : en amont du CAN, on place un FAR\footnote{Filte anti-repliement de spectre} , un filtre analogique passe-bas. + +\subsection*{Convertisseur Numérique Analogique} + +Problématique : transformer un échantillon numérique en signal analogique défini $\forall t$. + +Challenge théorique : quel comportement entre $(k-1)T_e$ et $kT_e$. + +Idée : extrapolation des échantillons entre 2 instants d'échantillonnage. + +\paragraph{Cas du Bloqueur d'Ordre Zéro} + +$B_0(p)$ fonction de transfert du filtre réalisant le BOZ. +\[b_0(t) = 1_0^+(t) - 1_0^+(t-T_e)\] +Donc par transformée de Laplace inverse, +\[B_0(p) = \frac{1}{p}-\frac{1}{p}e^{-T_ep} \] +\[ \boxed{B_0(p) = \frac{1-e^{-T_ep}}{p}} \] + +\chapter{Fonctions de transfert en z} + +Rappel : filtre analogique linéaire + +\[ e(t) \rightarrow \boxed{G(p)} \rightarrow s(t) \] +\[ s(t) = g(t)*e(t) \] +\[ S(p) = G(p) E(p) \] + + + +\section{Premières propriétés} + +\begin{thm} +Si on applique un échantillonnage en entrée de e(t), +\[s(t) = g(t) * e^*(t) \] +\begin{enumerate} +\item $s^*(t) = g^*(t) * e^*(t)$ où $g^*(t)$ est l’échantillonnage de $g(t)$ + +\item $s_n = g_n * e_n = \sum_ {k=0}^n g_{n-k}e_k$ + +\end{enumerate} +\end{thm} + +\begin{dem} +\begin{enumerate} +\item +\[g^*(t) * e^*(t) = \int_0^{\infty} g^*(t-\tau)e^*(\tau)d\tau = ... \] + +\item +\begin{align*} +s^*(t) & = s(t)\sum_{k=0}^{\infty}\delta_0(t-kT_e) = \sum_{k=0}^{\infty}s_k\delta_0(t-kT_e) +\intertext{Or, $ s(t) = \int_0^{\infty}g(t-\tau)e^*(\tau)d\tau $ avec $ e^*(t) = \sum_ {k=0}^{\infty}e_k\delta_0(t-kT_e)$} +\text{donc } s(t) & = \int_ 0^{\infty}g(t-\tau)(\sum_{k=0}^{\infty}e_k\delta_0(\tau-kT_e))d\tau \\ +& = \sum_ {k=0}^{\infty}e_k \int_0^{\infty} g(t-\tau) \delta_0(\tau-kT_e) d\tau \\ +& = \sum_ {k=0}^{\infty}e_k g(t-kT_e) \\ +s_n = s(nT_e) & = \sum_ {k=0}^{\infty}e_kg((n-k)T_e) = \sum_ {k=0}^{\infty}e_kg_{n-k} +\end{align*} + +\end{enumerate} +\end{dem} + +Application : Discrétisation d'un système analogique avec CNA + BOZ + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\sbEntree{E} + + +\sbBloc{cna}{CNA, BOZ}{E} +\sbRelier[$u_k$]{E}{cna} + +\sbBloc[3]{sys}{G(p)}{cna} +\sbRelier[$u(t)$]{cna}{sys} + +\sbBloc[3]{can}{CAN}{sys} +\sbRelier[$y(t)$]{sys}{can} + +\sbSortie[3]{S}{can} +\sbRelier[$y_k$]{can}{S} + + +\end{tikzpicture} +\caption{Discrétisation d'un système analogique} +\end{figure} + +Hypothèse : Synchronisation des convertisseurs + +\[ u_k \rightarrow \boxed{H(z)} \rightarrow y_k \] + +\subsection*{Fonction de transfert pour asservissement numérique} + +On cherche à déterminer la fonction de transfert $H(z)=\frac{Y(z)}{U^*(z)}$ de l'asservissement numérique suivant : + + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\sbEntree{E} + +\sbBloc{calc}{CNA}{E} +\sbRelier[$u_k$]{E}{calc} + +\sbBloc[3]{boz}{$B_0(p)$}{calc} +\sbRelier[$u^*(t)$]{calc}{boz} + +\sbBloc[3]{sys}{$G(p)$}{sys} +\sbRelier{boz}{sys} + +\sbBloc[3]{can}{CAN}{sys} +\sbRelier[$y(p)$]{sys}{can} + +\sbSortie[3]{S}{can} +\sbRelier[$y^*(t)$]{can}{S} + + +\end{tikzpicture} +\caption{Asservissement numérique} +\end{figure} + +\begin{thm} +\[ \boxed{H(z) = (1-z^{-1}) Z[^*L^{-1}[\frac{G(p)}{p}]]} \] +\end{thm} + +\begin{dem} +\begin{align*} +Y(p) & = B_0(p)G(p)U^*(p) \\ +& = (1 - e^{-T_ep})\frac{G(p)}{p}U^*(p) \\ +& = \frac{G(p)}{p}U^*(p)-\frac{G(p)}{p}U^*(p)e^{-T_ep} +\intertext{On pose $ \tilde{G}(p) = \frac{G(p)}{p} $ } +Y(p) & = \tilde{G}(p)U^*(p) - \tilde{G}(p)U^*(p)e^{-T_ep} +\end{align*} + +Avec $\tilde{Y}(p) = \tilde{G}(p)U^*(p)$, par transformation inverse de Laplace, +\begin{align*} +\tilde{y}(t) & = \tilde{g}(t)*u^*(t) \\ +\tilde{y}_n & = \tilde{g}_n*u_n \\ +\tilde{Y}(z) & = \tilde{G}(z)U(z)\\ +\text{Ainsi, } Y(z) & = \tilde{Y}(z) - z^{-1}\tilde{Y}(z) \\ +& = (1-z^{-1})\tilde{Y}(z) \\ +H(z) & = (1-z^{-1})\tilde{G}(z) \\ +H(z) & = (1-z^{-1})Z[^*L^{-1}[\tilde{G}(p)]] +\end{align*} +\end{dem} + +\medskip +\noindent Remarque : comment choisir $T_e$ ? Tout système physique peut être représenté par un filtre passe-bas : +\[ \boxed{ \text{Règle empirique : } 6f_c \leq f_e \leq 24f_c } \] + +\subsection*{Propriétés des systèmes discrétisés} + +\begin{enumerate} +\item Un système analogique linéaire reste linéaire après discrétisation. +\item L'ordre du système est conservé. +\item Les pôles du système discrétisé $p_d$ sont liés aux pôles du système analogique $p_c = e^{T_e p_c}$ (cela vient de $z=e^{T_ep}$). Attention, c'est faux pour les zéros ! +\item La discrétisation d'une association en série n'est pas identique à la mise en série des discrétisés. +\end{enumerate} + +\section{Obtention d'une fonction de transfert en z à partir d'une équation récurrente} + +\[a_ny_{k+n} + ... + a_1y_{k+1}+a_0y_k = b_mu_{k+m} + ... + b_1u_{k+1} + b_0u_k \] +\[ \text{ avec } a_i,b_j \in \mathbb{R}, a_n \neq 0 \] + +Par causalité, on a $n \geq m$. \\ + +Rappel : Théorème d'avance +\[ Z[u_{k+d|d\in\mathbb{N}^*}] = z^d U(z) - z^d \sum_{i=0}^{d-1}u_iz^{-i} \] + +On applique la transformée en $z$ à ($EQ_n$) = $a_ny_{k+n} + ... + a_1y_{k+1}+a_0y_k$ +\[ +\begin{array}{lcll} +a_n y_{k+n} & & a_nz^nY(z) & - a_n [y_0...y_{n-1}][z^n...z]^T \\ ++ a_{n-1} y_{k+N-1} & & + a_{n-1}z^{n-1}Y(z) & - a_{n-1} [0, y_0...y_{n-2}][z^n...z]^T \\ +... & \quad \rightarrow \quad TZ \quad \rightarrow \quad & ... & ... \\ ++ a_1 y_{k+1} & & + a_1z^{n-1}Y(z) & - a_1 [0, ...,0,y_0][z^n...z]^T \\ ++ a_0 y_k & & +a_0zY(z) \\ +\hline +TZ(EQ_n) & = & (\sum_{l=0}^na_lz^l)Y(z) & - CI_y(z) +\end{array} +\] + +On fait de même avec ($EQ_m$) = $b_mu_{k+m} + ... + b_1u_{k+1} + b_0u_k$. + +Conditions initiales données : les $y_k, k = 0,...,n-1$ et $u_k,k=0,...,m-1$ + +\begin{align*} +CIy(z) & = \sum_{j=0}^{n-1} ( \sum_{l=0}^j a_{n-l}y_{l-j}) z^{n-j} \\ +CIu(z) & = \sum_{j=0}^{m-1} ( \sum_{l=0}^j b_{m-l}u_{l-j}) z^{n-j} +\end{align*} + +Ainsi, en posant \[A(z) = \sum_{l=0}^n a_lz^l \text{ et } B(z) = \sum_{l=0}^m b_lz^l \] + +\begin{align*} +& A(z)Y(z) - CIy(z) = B(z)U(z) - CIu(z) \\ +& \boxed{Y(z) = \frac{B(z)}{A(z)} U(z) + \frac{CIy(z)-CIu(z)}{A(z)}} +\end{align*} + +On pose $G(z)=\frac{B(z)}{A(z)}$, appelée fonction de transfert du système. + +\begin{align*} +Y(z) & = G(z)U(z) + \frac{CI(z)}{A(z)} \\ +& \text{ où } CI(z) = CIy(z) - CIu(z) \\ +& \text{ À CI nulles, } Y(z) = G(z) U(z) +\end{align*} + +\subsection*{Définitions} +Les pôles (zéros) du système sont les racines de $A(z)$ ($B(z)$). + +Le gain statique (si défini) est \( \lim_{z\rightarrow 0}G(z)\). + +Lorsqu'il n'y a plus de simplifications possibles entre pôles et zéros dans G(z), on parle de fonction de transfert minimale. Alors, le degré de A(z) désigne l'ordre du système. + +\section{Réponse temporelle de système à temps discret} + +On considère le système à temps discret : +\[u_k \rightarrow \boxed{G(z)} \rightarrow y_k \] + +\[G(z) = \frac{B(z)}{A(z)} \text{ et } A(z) = \sum_{l=0}^n a_lz^l, B(z) = \sum_{l=0}^m b_lz^l\] + +\subsection{Calcul à partir de la relation de récurrence} + +On effectue un changement de variable muet pour exprimer $y_k$ en fonction des instants précédents. +\[a_ny_{k+n} + ... + a_1y_{k+1}+a_0y_k = b_mu_{k+m} + ... + b_1u_{k+1} + b_0u_k \] +\[a_ny_k = -a_{n-1}y_{k-1}-...-a_1y_{k-n+1}-a_0y_{k-n} + b_mu_{k+m-n} + ... + b_1u_{k-n+1} + b_0u_{k-n}\] +Intérêt : pratique pour le calcul en temps réel (simulation, implantation systèmes embarqués...). Les CI $y_{-1}, y_{-2}...$ sont à préciser + +\subsection{Calcul à partir de la fonction de transfert} +Si les CI sont nulles : +\begin{align*} +Y(z) & = G(z)U(z) \\ +y_k & = Z^{-1}[G(z)U(z)] +\end{align*} +En pratique, on effectue une décomposition en éléments simples de $\frac{Y(z)}{z}$ et on applique $Z^{-1}[.]$ à $Y(z)$ en utilisant le tableau des transformées en z usuelles.\\ + +Exemple : +\begin{align*} +\intertext{On cherche la réponse impulsionnelle ($u_k=\delta_k$) de } +G(z)& = \frac{1}{(z-1)(z-2)} +\intertext{On effectue la décomposition en éléments simples de $\frac{Y(z)}{z}$} +\frac{Y(z)}{z} & = \frac{1}{z(z-1)(z-2)} \\ +& = \frac{1}{2z} - \frac{1}{2(z-1)} + \frac{1}{2(z-2)} \\ +Y(z) & = \frac{1}{2} - \frac{1}{2}\frac{z}{z-1} + \frac{1}{2} \frac{z}{z-2} \\ +y_k &= \frac{1}{2}\delta_k + \frac{1}{2}2^k -1 +\end{align*} + +SI les CI non nulles et connues : +\begin{align*} +Y(z) & = G(z) U(z) + \frac{CIy(z)-CIu(z)}{A(z)} \\ +y_k &= Z^{-1}[G(z) U(z) + \frac{CIy(z)-CIu(z)}{A(z)}] +\end{align*} + +\subsection{Par décomposition modale} + +\begin{align*} +G(z) & = \frac{b_mz^m + ... + b_0}{a_nz^n + ... + a_0} = \frac{B(z)}{A(z)} \\ +& = \frac{K(z-z_1)^{\alpha_1}(z-z_2)^{\alpha_2}...(z-z_r)^{\alpha_r}}{(z-p_1)^{\gamma_1}(z-p_2)^{\gamma_2}...(z-p_q)^{\gamma_q}} \text{ avec } \sum_1^q \gamma_i = n, \sum_1^r \alpha_i = m +\end{align*} + +$\gamma_i$ est la multiplicité algébrique du pôle $p_i \in \mathbb{C}$. + +$\alpha_i$ est la multiplicité algébrique du zéro $z_i \in \mathbb{C}$ + +Avec l'hypothèse $a_n=1$, $A(z)$ est un polynôme appelé Monique. + +\begin{align*} +\frac{Y(z)}{z} & = \frac{G(z)U(z)}{z} \text{ où U(z) quelconque, de pôles } r_1,...,r_u \\ +\text{ d'où } Y(z) & = Y(0) + \sum_{i=1}^q G_i(z) + \sum_{i=1}^{r_u}U_i(z) +\end{align*} + +Remarque : $U(z)$ influence la décomposition de $G(z)$ et vice-verse. + +\begin{align*} +G_i(z) & = \sum_{j=1}^{\gamma_i}\frac{c_{ij}z}{(z-p_j)^j} \\ +g_{i_k} & = Z^{-1}[G_i(z)] \\ +& = (c_0 + c_1k + ... + c_{\gamma_i - 1}k^{\gamma_i - 1})p_i^k +& = P_i(k)p_i^k +\end{align*} + +$g_{i_k}$ correspond à l'évolution de la sortie $y_k$ due au pôle $p_i$ : mode $p_i$. + +\medskip + +La sortie $y_k$ est construite à partir de la contribution de chaque mode (et du type d'entrée) +\[ y_k = Y(0)\delta_k + \sum_{i=1}^q g_{i_k} + Z^{-1}[\sum_{i=1}^{r_u}U_i(z)] \] +où + +$\sum_{i=1}^q g_{i_k}$ est l'excitation des modes par l'entrée $y_k$ + +$Z^{-1}[\sum_{i=1}^{r_u}U_i(z)]$ le régime forcé par $u_k$ + +\subsubsection{Mode réel} + +\begin{itemize} +\item $|p_i|<1 \Rightarrow P_i(k)p_i^k \rightarrow_{\infty} 0$ : mode convergent +\item $|p_i|>1 \Rightarrow P_i(k)p_i^k $ divergence exponentielle +\item $ |p_i| = 1 et P_i(k) = c_0$ constant $\rightarrow$ mode entretenu (ni convergence, ni divergence) +\item $ |p_i| = 1 $ et $ \gamma_i > 1, P_i(k)p ^k \rightarrow $ divergence polynomiale +\begin{itemize} +\item Si $p_i > 0$ alors $ P_i(k)p_i^k$ tend à être du même signe : mode apériodique +\item Si $p_i < 0$ alors $P_i(k)p_i^k = (-1)^k |p_i|^k P_i(k)$ change de signe en fonction de la parité de k : mode oscillant +\item Si $p_i = 0 \rightarrow P_i(k)p_i^k = 0 \forall +k\geq 1$ : mode à réponse pile +\end{itemize} +\end{itemize} +Remarque : un pôle discret nul $p_i = 0$ possède un équivalent en temps continu à partie réelle infiniment négative : +\[p_i = e^{T_ep_{ci}} = 0 \Leftrightarrow p_{i} \rightarrow - \infty\] + +\subsubsection{Mode complexe} +À un pôle $p_i$ complexe correspond son conjugué $\overline{p_i}$ : +\[P_{a_i}(k)p_i^k + P_{b_i}(k)\overline{p_i}^k = ... = P(k)\rho_i^k\sin(k\theta_i+\phi\] +où $p_i = \rho_ie^{j\theta_i}$ et $\phi$ dépend du contexte. +\begin{itemize} +\item $|p_i| = \rho_i > 1$ : divergence +\item $|p_i| = \rho_i < 1$ : convergence en $\rho_i^k$ +\item $|p_i| = \rho_i = 1$ +\begin{itemize} +\item si multiplicité de $p_i = 1$ : mode entretenu +\item si multiplicité de $p_i > 1$ : divergence +\end{itemize} +\item $\theta_i \neq 0$ oscillation à la fréquence $\theta$ +\end{itemize} + +\section{Stabilité} + +\begin{dfn}[Stabilité EBSB] +Un système discret est stable au sens EBSB si pour toute entrée $u_k$ bornée, $y_k$ reste bornée. +\end{dfn} + +\begin{thm}[Stabilité et réponse impulsionnelle] +Un système est stable au sens EBSB si et seulement si sa réponse impulsionnelle est absolument sommable, c'est-à-dire $\sum_ {k=0}^{\infty}|g_k|<\infty$ +\end{thm} + + +\subsection*{Théorème : stabilité et pôles} +\begin{thm}[Stabilité et pôles] +Un système discret est stable au sens EBSB si et seulement si tous les pôles de sa fonction de transfert en $z$ sont à l'intérieur du cercle unité (strictement, pas sur le cercle). +\end{thm} + +Remarque : cela suppose le calcul des pôles de $G(z) = \frac{B(z)}{A(z)}$ + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.8]{polesenz.jpg} +\caption{Allure de la réponse temporelle en fonction de la position des pôles dans le plan $z$} +\end{figure} + +\subsection*{Critère de Jury} +\emph{À savoir utiliser, voir polycopié.} + +\subsection*{Critère de Routh-Hurwitz} +\emph{À connaître par coeur, voir polycopié.}\\ + +Rappel : En temps continu, le critère de Routh-Hurwitz permet de déterminer le nombre de racines instables de l'équation caractéristique, c'est-à-dire à partie réelle strictement positive.\\ + +Transformation en w : +\begin{align*} +z & = \frac{1+w}{1-w} ,\quad w \neq 1 \\ +w & = \frac{z-1}{z+1} ,\quad z \neq -1 +\end{align*} + +Cette transformation transforme le disque unité du plan en $z$, en le demi-plan ouvert gauche du plan en $w$. + +Cette transformation étant bijective, on l'utilise pour appliquer le critère de Routh au polynôme en la variable $w$. + +\subsection*{Critère de stabilité de Schur-Cohn} +\emph{Non exigible, voir polycopié.} + +\subsection*{Critère de stabilité de Nyquist} +\emph{À connaître par coeur, voir polycopié.} + +\chapter{Calcul de correcteurs en temps discret} + +\section{Le correcteur PID Proportionnel - Intégral - Dérivée} + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.5]{pid.png} +\caption{Correcteur PID} +\end{figure} + +Le PID se traduit donc théoriquement par : +\[ C(p) = K_p ( 1 + \frac{1}{T_ip} + T_dp ) \] + +En pratique, on considère plutôt : +\[ C(p) = K_p ( 1 + \frac{1}{T_ip} + \frac{T_dp}{1+\epsilon _p})\] +où $\epsilon < \frac{T_d}{10^N}$, avec $N>>10$ + +\begin{align*} +U(p) & = C(p) \epsilon(p) \\ +& = K_p \epsilon(p) + \frac{1}{T_ip} \epsilon(p) + T_dp\epsilon(p) \\ +u(t) & = K_p ( \epsilon (t) + \frac{1}{T_i} \int_0^t\epsilon(\tau)d\tau + T_d \frac{d\epsilon(t)}{dt} +\end{align*} + +Avec $t = kT_e, u_k = u(kT_e), \epsilon_k = \epsilon(kT_e)$ +\begin{align*} +u_k & = K_p \epsilon_k + \frac{K_p}{T_i} \int_0^{kT_e} \epsilon (\tau) d\tau + K_pT_d \frac{\epsilon_k - \epsilon_{k-1}}{T_e} \\ +& \text{ Or, } \int_0^{kT_e} \epsilon(\tau)d\tau \approx \sum_{j=0}^k \epsilon_jT_e \\ +u_k &= K_p \epsilon_k + \frac{K_pT_e}{T_i} \sum_{j=0}^k \epsilon_j + \frac{K_pT_e}{T_i}(\epsilon_k-\epsilon_{k-1}) \quad (1) +\end{align*} +où $K_p, T_i, T_d$ sont trois paramètres à ajuster pour garantir stabilité en performances en boucle fermée. + +Soit $U(z) = Z[u_k], \epsilon(z) = Z[\epsilon_k]$ +\begin{align*} +U(z) & = K_p ( \epsilon(z) + \frac{T_e}{T_i}\frac{z}{z-1}\epsilon(z) + \frac{T_d}{T_e}(1-z^{-1})\epsilon(z) ) \\ +\text{ d'où } C(z) & = \frac{U(z)}{\epsilon(z)} = K_p ( 1 + \frac{T_e}{T_i}\frac{z}{z-1} + \frac{T_d}{T_e}(1-z^{-1})) +\end{align*} + +Remarque : la loi de commande (1) n'est pas appropriée pour une implémentation en temps réel, car le terme $\sum_{j=0}^k \epsilon_j$ est difficile à manipuler. + +On s'intéresse donc plutôt à : + +\[ +u_k - u_{k-1} = K_p ( \epsilon_k - \epsilon_{k-1} ) + \frac{K_pT_e}{T_i} \epsilon_k + \frac{K_pT_d}{T_e}(\epsilon_k - 2\epsilon_{k-1} + \epsilon_{k-2}) +\] + +Méthodes de réglage du PID : +\begin{itemize} +\item méthodes graphiques, réglage des marges de gain et de phase, bande passante, ..., sur le diagramme de Bode +\item méthodes empiriques, mais la méthode de Takahashi\footnote{À savoir appliquer, voir polycopié} ne s'applique ni aux systèmes instables, ni aux systèmes oscillants. +\end{itemize} +\medskip + +Les effets des composantes P, I et D sont similaires à ceux dans le cas analogique. +\begin{itemize} +\item l'action P augmente la bande passante, donc la rapidité du système, mais dégrade la stabilité. Une amplitude de commande élevée conduit alors à une saturation. +\item l'action I augmente la précision, dégrade la rapidité et la stabilité. Elle est peu robuste aux perturbations basses fréquences (une perturbation constante sera intégrée). +\item l'action D augmente la rapidité et la bande passante. Elle permet d'améliorer les marges de stabilité mais amplifie les bruits hautes fréquences (notamment les bruits de capteurs) +\end{itemize} +\medskip + +Conclusion : il faudra combiner judicieusement ces actions en fonction du système, des capteurs et des performances requises. + +\section{Transposition des méthodes analogiques} + +\subsection{Approximation du BOZ par un retard équivalent} + +On rappelle l'expression de la fonction de transfert du BOZ : +\begin{align*} +B_0(p) & = \frac{1-e^{-T_ep}}{p} \\ +& = \frac{e^{-\frac{T_e}{2}p} ( e^{\frac{T_e}{2}p} - e^{-\frac{T_e}{2}p})}{p} \\ +\text{ Or, } & e^{-\frac{T_e}{2}p} = 1 - \frac{T_e}{2}p + o(\frac{T_e}{2}p) \\ +& e^{-\frac{T_e}{2}p} = 1 + \frac{T_e}{2}p + o(\frac{T_e}{2}p) \\ +\text{Donc } & \boxed{ B_0(p) \approx T_e e^{-\frac{T_e}{2}p}} +\end{align*} + +\subsection{Approximation de Padé pour les retards} + +Cette approximation repose sur le développement de Taylor du terme de retard exponentiel. Elle fournit une fraction rationnelle causale. + +À l'ordre 1, +\[ e^{-T_ep} = \frac{e^{-\frac{T_e}{2}p}}{e^{+\frac{T_e}{2}p}} = \frac{1 - \frac{T_e}{2}p}{1 + \frac{T_e}{2}p} \] + +À l'ordre 2, +\[ e^{-T_ep} = \frac{1 - \frac{T_e}{2}p + \frac{T_e^2}{8}p^2}{1 + \frac{T_e}{2}p + \frac{T_e^2}{8}p^2} \] + +Application au BOZ : +\[\boxed{B_0(p) \approx \frac{T_e}{1+\frac{T_e}{2}p}}\] + +Conséquence : on peut donc appliquer les résultats des systèmes analogiques sur le système équivalent obtenu. + +\subsection{Correction numérique obtenue par discrétisation approchée d'un correcteur continu} + +\subsubsection*{Approximation de l'opérateur intégral} + +\begin{align*} +x(t) & = \int_ 0^t e(\tau)d\tau\\ +x_k & = x(kT_e) = \int_ 0^{kT_e} e(\tau)d\tau \\ +x_k & = \sum_{j=0}^k e_jT_e +\end{align*} + + +\subsubsection*{Approximation d'Euler arrière} +\begin{align*} +\frac{de(t)}{dt} & = \frac{e_k - e_{k-1}}{T_e} \\ +pE(p) & = \frac{1-z^{-1}}{t_e}E(z) +\end{align*} +\[ \boxed{p = \frac{z-1}{zT_e}} \] + +\subsubsection*{Approximation d'Euler avant} +\begin{align*} +\frac{de(t)}{dt}& = \frac{e_{k+1}-e_k}{T_e} \\ +pE(p) & = \frac{z-1}{T_e}E(z) +\end{align*} +\[ \boxed{p = \frac{z-1}{T_e} } \] + + +\subsubsection*{Approximation de Tustin} +\begin{align*} +x_k - x_{k-1} & = \frac{1}{2}(e_k + e_{k-1})T_e \\ +(1-z^{-1})X(z) & = \frac{T_e}{2}(1+z^{-1})E(z) \\ +X(z) & = \frac{T_e}{2}\frac{1+z^{-1}}{1-z^{-1}}E(z) \\ +& = \frac{T_e}{2}\frac{z+1}{z-1}E(z) +\end{align*} + +D'où \[ \boxed{p = \frac{2}{T_e}\frac{z-1}{z+1}} \] + +Remarque : semblable à la transformation en $w$. + +Remarque : les approximations de $p$ induisent des distorsions fréquentielles. + +Exemple : correcteur continu $R_C(p)$ \\ +$R_d(z) = R_c(p) |_{p=\frac{2}{T_e} \frac{z-1}{z+1}}$ + +Réponse fréquentielle : $ z=e^{T_ep}, p=j\omega$ + +\begin{align*} +R_d(e^{jT_e\omega}) & = R_c(\frac{2}{T_e} \frac{jT_e\omega-1}{jT_e\omega+1} +\\ +\frac{jT_e\omega-1}{jT_e\omega+1} & = j\tan(\frac{T_e}{2}\omega) \\ +R_d(e^{jT_e\omega}) & = R_c(j\frac{2}{T_e} \tan(\frac{T_e}{2})\omega) +\\ +& = R_c(j\tilde{\omega}) \text{ où } \tilde{\omega} = \frac{2}{T_e}\tan(\frac{T_e}{2}\omega) +\end{align*} + +$\tilde{\omega}$ est une "pseudo-pulsation" qui varie de 0 à $+\infty$ lorsque $\omega$ varie de 0 à $\frac{\pi}{2}$. Cela correspond à une distorsion de l'échelle fréquentielle. + +Approximation de Tustin adaptée à la pulsation $\omega_c$ + +On voudrait que $R_d(e^{j\omega_cT_e}) = R_c(j\omega_c)$, alors +\[p \leftarrow \frac{\omega_c}{\tan(\frac{\omega_cT_e}{2})}\frac{z-1}{z+1}\] +\[R_c(j\frac{\omega_c}{\tan(\frac{\omega_cT_e}{2}})) = R_c(j\omega_c)\] + +Approximation par correspondance pôle-zéro + +Exemple : +\begin{align*} +R_c(p) & = \frac{p+a}{p+b} \\ +z & = e^{T_ep} \\ +R_d(z) & = \frac{z-e^{-T_ea}}{z-e^{-T_eb}} \alpha \\ +\text{Gain statique : } & \lim_{z\rightarrow1}R_d(z) = \alpha \frac{1-e^{-T_ea}}{1-e^{-T_eb}} = \lim_ {p\rightarrow 0} R_c(p) = \frac{a}{b} \\ +\alpha & = \frac{a}{b} \frac{1-e^{-T_ea}}{1-e^{-T_eb}} +\end{align*} + +En résumé, on construit $R_d(z)$ avec la même structure que $R_c(p)$ en temres de zéros, pôles et gain statique. +Précaution à prendre lorsque le degré du numérateur de $R_c(p)$ est inférieur au degré du dénominateur de $R_c(p)$ (i.e. $R_c(p)$ strictement propre) + +Exemple : +\begin{align*} +R_c(p) & = \frac{p+a}{(p+b)(p+c)} \\ +R_d(p) & = \frac{(z+1)(z - e^{-T_ea})}{(z-e^{-T_eb)(z-e^{-T_ec})}}\alpha \\ +R_d(1) = R_c(0) \rightarrow \alpha = ... +\end{align*} +Le terme $(z+1)$ est ajouté pour permettre d'avoir le même gain de $R_d(z)|_{z=e^{jT_e\frac{\pi}{T_e}}} = R_c(j\omega)|_{\omega \rightarrow \infty}$ (correspondance du gain haute fréquence). + +En conclusion, le choix d'une approximation dépend beaucoup des caractéristiques (zéros, ordre,...) du système. + +\newpage +\textbf{À savoir pour le partiel} +\begin{itemize} +\item Transformées en $z$ et ses propriétés +\item Tous les résultats sur les filtres linéaires numériques +\item Méthode d'obtention des transformées en $z$ d'un système discrétisé +\item Définitions stabilité, critères algébriques pour la caractériser (connaître Routh-Hurwitz, savoir appliquer Jury) +\item Réglage d'un correcteur PID analogique, savoir passer en TD (Euler, Tustin, méthode de correspondance pour le zéro) +\item Transformation en $w$ +\item Correcteur RST +\item Règle des retards relatifs +\item Règles de rejet de la perturbation, conséquence sur le polynôme $R$ +\item Être en mesure de déterminer les degrés de polynômes R et S pour résoudre un problème : technique de simplification de pôles / zéros, introduction de polynôme auxiliaire +\end{itemize} + +\chapter{Commande dans l'espace d'état} + +\section{Concept du modèle d'état} + +\subsection{Définitions} +Soit un système $\Sigma$, à temps continu, linéaire ou non : +\[ u \rightarrow \boxed{\Sigma} \rightarrow y\] + +$u(t) \in \R^m$ commande + +$y(t)\in \R^p $ sortie mesurée + +$x(t) \in \R^n$ vecteur d'état, et ses composantes $x_i(t)\in \mathbb{R}$ variables d'état\\ + +On appelle équation d'état du système $(\Sigma)$ : +\[ \dot{x}(t) = f(x(t),u(t),t) \] + +$f : \R^n \times \R^m \times R_+ \rightarrow \R^n$ : champs de vecteurs, relation non linéaire en $x,u,t$. + +$x_0=x(0) \in \R^n$ : vecteur des conditions initiales. + +$x(t)$ contient des grandeurs physiques ou non.\\ + +On appelle équation d'observation du système $(\Sigma)$ : +\[ y(t) = h(x(t),u(t),t) \quad \text{(équation algébrique)} \] + +Un modèle d'état est composé d'une équation d'état et d'une équation d'observation : + +\[ +(\Sigma) +\left\{ +\begin{array}{ll} +\dot{x}(t) & = f(x(t),u(t),t), \quad x_0=x(0) \\ +y(t) & = h(x(t),u(t),t) +\end{array} +\right. +\] + +Dans le cas discret : + +\[ +(\Sigma) +\left\{ +\begin{array}{ll} +x_{k+1} & = f_d(x_k,u_k,k), \quad x_0=x(0) \\ +y_k & = h_d(x_k,u_k,k) +\end{array} +\right. +\] + +\[ u_k\in \R^m \rightarrow \boxed{\Sigma_d} \rightarrow y_k\in\R^p \] + +$x_k \in \R^n$ vecteur de suites numériques + +\paragraph{Systèmes stationnaires :} on peut simplifier comme suit : + +\begin{multicols}{2} +Continu : +\[ +\left\{ +\begin{array}{ll} +\dot{x}(t) & = f(x(t),u(t)), \quad x_0=x(0) \\ +y(t) & = h(x(t),u(t)) +\end{array} +\right. +\] + +Discret : +\[ +\left\{ +\begin{array}{ll} +x_{k+1} & = f_d(x_k,u_k), \quad x_0=x(0) \\ +y_k & = h_d(x_k,u_k) +\end{array} +\right. +\] +\end{multicols} + + +\paragraph{Systèmes linéaires stationnaires :} on a alors : + \begin{multicols}{2} +Continu : +\[ +\left\{ +\begin{array}{ll} +\dot{x}(t) & = Ax(t)+Bu(t), \quad x_0=x(0) \\ +y(t) & = Cx(t)+Du(t)) +\end{array} +\right. +\] + +Discret : +\[ +\left\{ +\begin{array}{ll} +x_{k+1} & = A_d x_k + B_d u_k, \quad x_0=x(0) \\ +y_k & = C_d x_k + D_d u_k +\end{array} +\right. +\] +\end{multicols} + +$A,A_d \in \R^{n\times n}$ matrices d'évolution + +$B,B_d\in \R^{n\times m}$ matrices d'application de l'entrée commande $u$ + +$C,C_d \in \R^{p \times n}$ matrices d’observation + +$D,D_d \in \R^{p\times n}$ matrices de transmission directe + +\paragraph{Remarque} Systèmes linéaires variant dans le temps : + +\begin{multicols}{2} +Continu : +\[ +\left\{ +\begin{array}{ll} +\dot{x}(t) & = A(t)x(t)+B(t)u(t), \quad x_0=x(0) \\ +y(t) & = C(t)x(t)+D(t)u(t)) +\end{array} +\right. +\] + +Discret : +\[ +\left\{ +\begin{array}{ll} +x_{k+1} & = A_d(k) x_k + B_d(k) u_k, \quad x_0=x(0) \\ +y_k & = C_d(k) x_k + D_d(k) u_k +\end{array} +\right. +\] +\end{multicols} + +\paragraph{Schéma-bloc :} cas linéaire stationnaire + +Insérer schéma bloc + + +\paragraph{Cadre du cours :} systèmes linéaires, stationnaires, SISO (mono-entrée, mono-sortie). + +De plus, $u\in\R \quad (m=1)$, $y\in\R \quad (p=1)$. + + +\subsection{Quelques exemples} + +\subsubsection{Robot manipulateur à 1 bras} + +PFD : équation de mouvement + +\begin{align*} +J\ddot{\theta}(t) & = C_m(t) + C_g(t)\\ +J\ddot{\theta} & = C_m(t) - \frac{L}{2}mg\cos\theta \\ +\intertext{On pose $C_m(t) = u(t)$} +\ddot{\theta}(t) & = \frac{1}{J}u(t) - \frac{mgL}{2J}\cos(\theta(t)) +\end{align*} +On a donc : +\[ +\dot{x}(t) = +\left[ +\begin{array}{cc} +\dot{\theta}(t) \\ +\ddot{\theta}(t) +\end{array} +\right] += +\left[ +\begin{array}{cc} +\dot{\theta}(t) \\ +-\frac{mgL}{2J}\cos(\theta(t)) +\end{array} +\right] ++ +\left[ +\begin{array}{cc} +0 \\ \frac{1}{J} +\end{array} +\right] u(t) +\] + +\subsubsection{Cas d'un robot à n liaisons en série, n-actionné} + +\[M(q) \ddot{q} + C(q,\dot{q}) \dot{q} + D\dot{q} + g(q) = \tau \] +\[q(t) = \vect{ +\theta_1(t) \\ +\vdots \\ +\theta_n(t) +} +\in \R^n, \quad +\tau = +\vect{ +\tau_1(t) \\ +\vdots \\ +\tau_n(t) +}\] + +\begin{tabular}{lll} +$M(q) = M(q)^T$ & $\in \R^{n \times n}$ & matrice d'inertie, définie positive \\ +$C(q,\dot{q})\dot{q}$ & $\in \R^n$ & forces centrifuges et de Coriolis\\ +$g(q) = \frac{\partial u(q)}{\partial q}$ & $\in \R^n$ & énergie potentielle totale due à la gravité\\ +$D\dot{q}$ & & frottements visqueux dans les liaisons +\end{tabular} + + +\[x(t) = +\vect{q(t) \\ \dot{q}(t)} \in \R^{2n}\] +\[\dot{x}(t) = +\vect{ \dot{q}(t) \\ \ddot{q}(t)} += +\vect{\dot{q}(t) \\ -M^{-1}(q)C(q,\dot{q})\dot{q}-M^{-1}D\dot{q}-M^{-1}(q)g(q)} ++ +\vect{ 0_{n\times n} \\ M^{-1}} +\] + +\subsubsection{Circuit avec diode à effet tunnel} + +\begin{figure} +\centering +\begin{tikzpicture}[scale=0.8] +\draw (0,0) to [sV=$u(t)$] (0,2) to [R,l=$R$] (0,4) to [L,l=$L$,i=$i_L$] (4,4) to [C,l=$C$,v=$v_c$] (4,0) to (0,0) +(4,4) to (8,4) to [tunnel diode,i=$i_D$] (8,0) to (4,0) +; +\end{tikzpicture} +\end{figure} + +Lois de Kirchoff : +\begin{align*} +\text{Noeud A : } & i_L = i_C + i_D = h(v_D) + v_C \\ +& \Rightarrow C\frac{dv_D}{dt} = i_L - h(v_D)\\ +\text{Maille 1 : } & u(t) - Ri_L(t) - L\frac{di_L}{dt} -v_C(t) = 0 \\ +& \Rightarrow Ri_L + L\frac{di_L}{dt} + v_D = u(t)\\ +\text{Maille 2 : } & v_C(t) = v_D(t) +\end{align*} +\[x(t) = \vect{ +i_L \\ +v_D +} = \vect{ +x_1 \\ +x_2 +} \in \R^2,\text{ 2 variables d'état} \] + +\[\dot{x}(t) = +\vect{ +\dot{x}_1 \\ +\dot{x}_2 +} + = +\left[ +\begin{array}{cc} +-\frac{R}{L}i_L & -\frac{1}{L}v_D \\ +\frac{1}{C}i_L & -\frac{1}{C}h(v_D) +\end{array} +\right] + + + \vect{ + \frac{1}{L} \\ + 0 + } + u(t) + \] + + +\subsection{Quelques propriétés de base, valables à temps continu ou discret} + +\subsubsection{Non-unicité d'un modèle d'état} + +Cas continu : +\[ +\left\{ +\begin{array}{ll} +\dot{x}(t) & = Ax(t)+Bu(t), \quad x_0=x(0) \\ +y(t) & = Cx(t)+Du(t)) +\end{array} +\right. +\] + +Soit $T\in\R^{n\times n}$ inversible. + +Soit $z(t) \in \R^n$ tel que + +\begin{align*} +x(t) & = Tz(t) \\ +\dot{x}(t) & = T\dot{z}(t) +\end{align*} +\begin{align*} +& \left\{ +\begin{array}{ll} +T\dot{z}(t) & = AT(t)+Bu(t) \\ +y(t) & = CTz(t)+Du(t)) +\end{array} +\right. \\ +& \left\{ +\begin{array}{ll} +\dot{z}(t) & = T^{-1}ATz(t) + T^{-1}Bu(t) \\ +y & = CT z(t) + Du(t) +\end{array} +\right., z(0) = T^{-1}x_0 \\ +& \left\{ +\begin{array}{ll} +\dot{z}(t) & = \tilde{A}z(t) + \tilde{B}z(t) \\ +y & = \tilde{C}z(t) + Du(t) +\end{array} +\right. +\end{align*} + +D est invariant par changement de coordonnées régulier ($x(t) = Tz(t)$) + +Il existe une infinité de modèles d'état pour un même système (linéaire et stationnaire). + +\paragraph{Remarque} voir plus bas quelques changement de coordonnées vers des formes d'état canoniques. + +\subsection{Modèle d'état pour quelques associations de systèmes (TD1)} + +Soient +\begin{align*} +(S_1) & \left\{ +\begin{array}{ll} +\dot{x}_1 & = A_1x_1 + B_1u_1 \\ +y_1 & = C_1 x_1 + D_1 u_1 +\end{array} +\right. \\ +(S_2) & \left\{ +\begin{array}{ll} +\dot{x}_2 & = A_2x_2 + B_2u_2 \\ +y_2 & = C_2 x_2 + D_2 u_2 +\end{array} +\right. +\end{align*} + +Association en série (schéma) + +\section{Solution de l'équation d'état} + +\subsection{Exponentiel d'une matrice} + +\begin{dfn} +Soit $\K = \R \ou \C$. +\[ \forall A \in \K^{n\times n}, e^A \in \K^{n\times n} \et e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!} \] +On admet la convergence de la série. +\end{dfn} + +\paragraph{Remarque} $e^{0_{n\times n}} = 1_{n}$ + +\begin{prop} +\[ e^A = \lim_{k \rightarrow \infty} (1_{n}+\frac{1}{k}A)^k \] +\begin{enumerate} +\item $(e^A)^T = e^{A^T}$ +\item $e^A$ inversible et $(e^A)^{-1} = e^{-A}$ +\item Si $A=diag(A_k)$ où $A_k\in \K^{n_i \times n_i}$, i=1.. k, $e^A = diag (e^{A_k})$ +\item Soit $X$ inversible $\in \K^{n \times n}$, $e^{XAX^{-1}} = Xe^AX^{-1}$ +\item Si $A,B \in \K^{n \times n}$ sont similaires $e^A$ et $e^B$ sont similaires aussi +\item Si $A,B \in \K^{n \times n}$ sont similaires et unitaires $e^A$ et $e^B$ le sont aussi +\item Si $A$ est hermitienne ($A=\overline{A}^T$) alors $e^A$ est définie positive +\item Si $A$ est anti-hermitienne, alors $e^A$ est unitaire +\item Si $A$ est normale $(AA^* = A^*A)$, alors $e^A$ est normale aussi +\end{enumerate} +\end{prop} + +\begin{prop} +\[ t\in \R, \frac{de^{At}}{dt} = Ae^{tA} = e^{tA}A \] +\end{prop} + +\begin{proof} +\begin{align*} +e^{tA} & = \sum_{k=0}^{\infty} \frac{t^k}{k!}A^k \\ +\frac{de^{At}}{dt} & = \frac{d}{dt}\sum_{k=0}^{\infty} \frac{t^k}{k!}A^k = \sum_{k=1}^{\infty} \frac{t^{k-1}}{(k-1)!}A^k \\ +& = A \sum_{k=1}^{\infty} \frac{t^{k-1}}{(k-1)!}A^{k-1} = Ae^{tA} +\end{align*} +\end{proof} + +\begin{prop} +Soient $A,B \in \K^{n \times n}$ +\begin{enumerate} +\item \[\forall t \in \R_+, \quad AB = BA \Leftrightarrow e^{tA}e^{tB} = e^{t(A+B)} \] +\item \[ \si AB = BA, \alors e^{A+B} = e^A e^B = e^Be^A \] +\end{enumerate} +\end{prop} + + +\paragraph{Théorème} Soit $A\in \K^{n\times n}$ + +\begin{align*} +e^{tA} & = \sum_{k=0}^{n-1} \Psi_k(t)A^k, \quad \forall t \in \R \\ +e^{tA} & = \frac{1}{2\pi j} \int_C (z1_n-A)^{-1} e^{tz} dz +\intertext{où $C$ est un contour fermé du plan complexe contenant $Spec(A)$ (valeurs propres de A)} +\intertext{Soit $P_A$ le polynôme caractéristique de $A$} +P_A(s) & = det(s1_n-A) = s^n + a_{n-1}s^{n-1} + ... + a_1s + a_0, \quad a_j \in \R, j=0...n-1 \\ +\intertext{On montre que} +sP_A^{(k+1)}(s) & = P_A^{(k)}(s) - a_k, \quad k=0,...n-1 \avec P_A^{(0)}(s) = P_A(s) \et P_A^{(n)}(s) = 1 \\ +\Psi_k(t) & = \frac{1}{2\pi j} \int_C \frac{P_A^{(k+1)}(z)}{P_A^{(k)}(z)} e^{tz} dz +\end{align*} + +On montre que $\forall k = 0,1...n-1$ et $t\geq 0$ +\[ \Psi_k^{(n)}(t) + a_{n-1} \Psi_k^{(n-1)}(t) + ... + a_1\Psi_k^{(0)}(t) + a_0\Psi_k(t) = 0 \] +avec $\forall k,l= 0...n-1, k\neq l$, on a $\Psi_k^{(l)}(0) = \delta_{kl}$ + +\paragraph{Théorème} Soit $A\in \K^{n\times n}$ et $\Psi_0(t)...\Psi_{n-1}(t)$ définis précédemment. + +Alors $\forall s \in \C \setminus Spec(A)$, +\[ L[e^{tA}] = \int_0^{\infty} e^{-st}e^{At} dt = (s1_n-A)^{-1} \] + +On appelle $(s1_n-A)^{-1}$ résolvante de A. + +De plus, +\begin{align*} +\hat{\Psi_k}(s) & = L[\Psi_k(t)]_{k=0..n-1} \\ +& = \frac{P_A^{(k+1)}(s)}{P_A^{(k)}(s)} \\ +(p1_n-A)^{-1} & = L[e^{tA}] \\ +& = \sum_{k=0}^{n-1}\hat{\Psi_k}(s)A^k +\end{align*} + +\subsection{Cas analogique} + +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x} &= Ax + Bu, \quad x(0)=x_0 \in \R^n \\ +y & = Cx+Du +\end{array} +\right. +\] + +\[ x(t) = e^{At}x_0 + \int_0^t e^{A(t-\tau)}Bu(\tau)d\tau \] + +\begin{proof} +\begin{align*} +\dot{x} & = Ax + Bu \\ +e^{-tA}(\dot{x} - Ax) & = e^{-tA}Bu\\ +\int \frac{d}{dt}(e^{-tA}x(t)) & = \int e^{-tA}Bu(t) \\ +e^{-tA}x(t)-e^{-t_0A}x_0 & = \int_{t_0}^t e^{-\tau A}Bu(\tau)d\tau \avec t_0=0 +\end{align*} + +Réciproquement +\begin{align*} +\dot{x}(t) & = Ae^{tA}x_0 + \frac{d}{dt} \int_0^t e^{A(t-\tau)}Bu(\tau)d\tau \\ +& = Ae^{tA}x_0 + \int_0^t Ae^{(t-\tau)A} Bu(\tau)d\tau + Bu(t) \\ +& = A(e^{tA}x_0 + \int_0^t e^{(t-\tau)A} Bu(\tau)d\tau) + Bu(t) \\ +& = Ax(t) + Bu(t) +\end{align*} +\end{proof} + +\subsection{Cas discret} +\[ (S) : +\left\{ +\begin{array}{ll} +x_{k+1} &= A_dx_k + B_du_k, \quad x_0 \in \R^n \\ +y_k & = C_dx_k+D_du_k +\end{array} +\right. +\] + +\[ +\begin{array}{lll} +k=0 & x_1 = & A_dx_0 + B_du_0 \\ +k=1 & x_2 = & A_dx_1 + B_du_1 \\ +& & A_d^2x_0 + A_dB_du_0 + B_du_1 \\ +& \vdots \\ +& x_k & = A_d^k x_0 + \sum_{j=0}^{k-1}A_j^{k-1-j}B_du_j +\end{array} +\] + +En pratique, soit $V\in \mathbb{K}^{n\times }$ inversible, tel que $V^{-1}AV = J$, $J\in \mathbb{K}^{n \times n}$ matrice de Jordan ou bien $J = \Lambda = \text{diag}(\lambda_1...\lambda_n), \lambda_i$ valeurs propres de A + +\begin{prop} +\[A^k = V J^k V^{-1}\] ou bien si $J = \Lambda = \text{diag}(\lambda_1...\lambda_n)$, \[A^k = V \Lambda^k V^{-1} \] + +\[ e^{tA} = e^{tVJV^{-1}} = V e^{tJ} V^{-1} \] +ou si $J = \Lambda = \text{diag}(\lambda_1...\lambda_n)$, +\[ e^{tA} = V e^{t\Lambda}V^{-1} \] + +$\Lambda^k = \text{diag}(\lambda_i^k)$ + +$e^{t\Lambda} = \text{diag}(e^{t\lambda_i})$ +\end{prop} + +Remarque : si $x(t)$ ou $x_k$ connu, +\[y(t) = Cx(t) + Du(t) \] +\[y-k = C_dx_k + d_du_k \] + +\section{Commandabilité et observabilité} + +Problème : existe-t-il une commande $u(t)$ permettant de passer d'un point de fonctionnement à $t=t_1$ à un autre à $t=t_2$ ? + +\subsection{Commandabilité} + +\begin{dfn}[Cas analogique] +Le système $(S)$ est dit commandable si \[\forall x(t=t_0)=x_0 \in \K^N \et \forall x_f=x(t=t_f)\in \K^n\] il existe une commande $u(t)$ continue (par morceaux) qui amène l'état $x(t)$ de l'état $x_0$ à $t=t_0$ vers $x_f$ à $t=t_f$. +\end{dfn} + +\begin{dfn}[Cas discret] +Le système $(S_d)$ est commandable si \[\forall x_d \in \K^n \et \forall x_f \in \K^n\] il existe une séquence d'échantillons de commande $[u_0,u_1,\dots u_k]$ qui amène le système $S_d$ de l'état de $x_d$ pour $k=0$ à $x_f$ pour $k=n$. +\end{dfn} + +\begin{dfn}[analogique ou discret] +On appelle matrice de commandabilité (dite de Kalman), la matrice notée (obtenue par concaténation) +\[ C(A,B) = [B \quad AB \quad A^2B \dots A^{n-1}B] \in \K^{n \times n} \] +\end{dfn} + +\begin{thm} Cas analogique ou discret : \\ + +Le système $(S)$ est commandable si et seulement si (matrice de rang plein) +\[ rang( C(A,B) = n )\] + +Le système $(S_d)$ est commandable si et seulement si +\[ rang( C(A_d,B_d) = n )\] + +\end{thm} + +\begin{prop}[Corollaire spécifique aux systèmes monovariables] +Le systèmes $(S)$ ou $(S_d)$ est commandable si et seulement si +\[ det (C(A,B)) \neq 0 \quad (\ou det (C(A_d,B_d)) \neq 0) \] +\end{prop} + +\begin{proof} +\[x_k= A_d^k x_0 + \sum_{j=0}^{k-1}A_j^{k-1-j}B_du_j\] +Pour atteindre n'impotre quel état de $\K^N$, il faut que +\[ Im\{A_d^{k-1-j}B_d\}_{j=0...k-1} = \R^n \] + +De plus, $\forall k \geq n$, +\[ Im\{A_d^{k-1-j}B_d\}_{j=0...k-1} = Im\{A_d^{n-1-j}B_d\}_{j=0...n-1} \] +En effet, d'après le théorème d'Hamilton-Cayley, la matrice $A$ est racine de son polynôme caractéristique : $P_A(A) = 0$ donc $A^n = -\sum_{k=0}^{n-1}a_k A^k$. + +Dire que $Im\{A_d^{k-1-j}B_d\}_{j=0...n-1} = \K^n \Leftrightarrow rang(C(A_d,B_d)) = n$ +\end{proof} + +\begin{prop} +Soit $x_d\in\K^n, x_f\in\K^n$, +\[x_f-A_d^nx_d = [B \quad AB \dots \quad A^{n-1}b][u_{n-1} \dots u_0]^T\] + +Si $C(A,B)$ inversible i.e. système commandable, alors on en déduit la séquence de commande permettant de passe de $x_d$ pour $k=0$ à $x_f$ pour $k=n$. +\end{prop} + +\subsection{Observabilité} + +\begin{dfn} +Le système $(S)$ est observable si $\forall x_0 \in \R^n$ pour $t=t_0$, il est possible de déterminer cet état uniquement en se servant de l'entrée $u(t)$ ou $u_k$ et de la sortie $y(t)$ ou $y_k$. +\end{dfn} + +\begin{dfn} +On appelle matrice d'observabilité (dite de Kalman), la matrice +\[ O(A,C) = \left[ \begin{array}{c} +C \\ +CA \\ +\vdots \\ +CA^{n-1} +\end{array} \right] +\in \K^{n\times n} +\] +\end{dfn} + +\begin{thm} +Le systèmes $(S)$ ou $(S_d)$ est observalble si et seulement si +\[ rang(O(A,C)) = n \quad (ou rang(O(A_d,C_d)) = n \] +\end{thm} + +\begin{prop}[Corollaire] +\[rang(O(A,C)) = n \Leftrightarrow det(O(A,C)) \neq 0 \] +\end{prop} + +\begin{proof} +Dans le cas discret, \[y_k = C_dA_d^kx_0 + \sum_{j=0}^{k-1}C_dA_d^{k-1-j}B_du_j+D_du_k\] +\[ \left[ \begin{array}{c} C \\ CA \\ \vdots \\CA^{k-1} \\ \end{array} \right] x_d = \left[ \begin{array}{c} y_0 \\ y_1 \\ \vdots \\ y_k \end{array} \right] - M \left[ \begin{array}{c} u_0 \\ u_1 \\ \vdots \\ u_{k-1} \end{array} \right] \] +avec \[ M = \left[ +\begin{array}{ccccc} +0 \\ +CB & \ddots\\ +CAB & \ddots & \ddots \\ +\vdots & \ddots & \ddots & \ddots \\ +CA^{k-2} & \hdots & CAB & CB & 0 +\end{array} +\right] +\] + +$k\geq n-1$ + +$x_d$ s'obtient si et seulement si $rang(O(C,A))=n$ +\end{proof} + + +\begin{dfn} +On définit le Gramien de commandabilité, noté $W_c\in\R^{n\times n}$ +\[W_c(t_0,t_1) = \int_{t_0}^t e^{\tau A} BB^T e^{\tau A^T} d\tau = W_c(t_0,t_1)^T \geq 0 \] +\end{dfn} + +\begin{thm} +Le système $(S)$ d'équation d'état $\dot{x}=Ax+Bu$ est commandable si et seulement si $W_c$ inversible, c'est-à-dire $W_c > 0$. +\end{thm} + +\begin{proof} +a) Si $W_c$ inversible, alors $\dot{x}=Ax+Bu$ (commandable) + +Soient $x_0,X_1\in\R^n$. + +Soit $v(t)$ bornée sur $[t_0,t_1]$ défini par $v(t) = B^Te^{A(t_1-\tau)}W_c(t_0,t_1)^{-1}(x_1-e^{A(t_1-t_0)}x_0)$, + +$v(t)$ solution de $\dot{\overline{x}} = A\overline{x}+Bv,\overline{x}(t_0) = X_0$ + +On a \begin{align*} +\overline{x}(t_1) & = e^{A(t_1-t_0)}x_0 + \int_{t_0}^{t_1}e^{A(t_1-\tau)}Bv(\tau)d\tau \\ +& = e^{A(t_1-t_0)}x_0 + \int_{t_0}^{t_1}e^{A(t_1-\tau)}BBe^{A^T(t_1-\tau)}W_c(t_0,t_1)^{-1}(x_1-e^{A(t_1-t_0)}x_0)d\tau + W_c(t_0,t_1)W_c(t_0,t_1)^{-1}(x_1-e^{A(t_1-t_0)}x_0), \forall x_1 \in \R^n \\ +& = x_1 +\end{align*} + +b) Si $\dot{x}=Ax+Bu$ commandable, alors $W_c$ inversible. Montrons que si $W_c$ non inversible, alors $\dot{x}=Ax+Bu$ non commandable. + +$\exists y \in \R^n \setminus \{0\}$ tel que $W_cy=0$ +\begin{align*} +\Leftrightarrow & y^TW_c y = 0 \\ +\Leftrightarrow & \int_{t_0}^{t_1}y^T e^{A\tau}BB^Te^{A^T\tau}yd\tau = 0 \\ +\Leftrightarrow & = 0 +\Rightarrow & B^Te^{A^T\tau}y=0 \forall\tau\in [t_0,t_1] \\ +\Leftrightarrow & y^Te^{A\tau}B=0 +\end{align*} + +Soit $u$ tel que $\dot{x}=Ax+Bu$, $x(t_0)=0$ +\[x(t_1) = \int_{t_0}^{t_1}e^{A(t_1-\tau)}Bu(\tau)d\tau \] +\[ \Rightarrow y^Tx(t_1)=0 \] + +Or si $y\in\R^n \setminus \{0\}$ il existe $x_1\in\R^n$ tel que $y^Tx_1\neq 0$ par exemple $x_1=y$ donc $\forall u, x(t_1) \neq x_1$ non commandable. +\end{proof} + +\begin{prop}[asymptotique du Gramien] +$W_c(0,\infty)$ solution de l'équation de Lyapumov $AP + PA^T+BB^T=0$ + +Gramien d'observabilité : +\[W_o(t_0,t_1) = \int_{t_0}^{t_1}e^{A^Tt}C^TCe^{At}dt \] +$W_o(0,\infty)$ solution de $A^TP + PA + C^TC = 0$ +\end{prop} + +\section{Stabilité} + +\emph{Voir polycopié} + +\section{Relation modèle d'état / fonction de transfert} + +\subsection{Modèle d'état vers fonction de transfert} + +\[ u(t) \rightarrow \boxed{(S)} \rightarrow y(t) \] +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x} &= Ax + Bu, \quad x(0)=x_0 \in \R^n \\ +y & = Cx+Du +\end{array} +\right. +\] + +Soient $Y(p) = L\{y(t)\}, U(p) = L\{u(t)\}, p\in \C$. $X(p) = L\{x(t)\} = [X_1(p) \dots X_n(p)]^T$ + +Alors on a +\begin{align*} +L\{\dot{x}\} & = pX(p) - x_0 \\ +pX(p) - x_0 & = AX(p) + BU(p) \\ +(p1_n-A)X(p) & = BU(p) + x_0 \\ +X(p) & = (p1_n-A)^{-1}BU(p) + (p1_n-A)^{-1}x_0 \\ +\intertext{ Remarque : $X(p) = L\{\int_0^t e^{-A(t-\tau)}Bu(\tau)d\tau\} + L\{e^{At}x_0\}$} +Y(p) & = CX(p) + DU(p) \\ +Y(p) & = [C(p1_n-A)^{-1}B+D]U(p) + C(p1_n-A)^{-1}x_0 + DU(p) +\end{align*} + +Soit $G(p)$ la fonction de transfert entre $u$ et $y$. +\[\boxed{G(p) = C(p1_n-A)^{-1}B+D}\] + +\begin{prop} +Les valeurs propres de A (les modes de (S)) sont les pôles de la fonction de transfert $G(p)$. +\end{prop} + +\begin{proof} +\[(p1_n-A)^{-1}=\frac{1}{det(p1_n-A)}Adj(p1_n-A)\] +Or, $P_A(p) = det(p1_n-A)$. $Adj(p1_n-A)\in\K^{n\times n}[X]$ Les éléments de $Adj(p1_n-A)$ sont des polynômes d'ordre $n-1$ +\[G(p) = \frac{CAdj(p1_n-A)B+DP_A(p)}{P_A(p)}\] +\end{proof} + +\subsection{Fonction de transfert / équation différentielle vers modèle d'état} + +\emph{Voir polycopié} + +\subsubsection*{Formes canoniques à matrice d'évolution compagnon} +\paragraph{Exemple :} $4y^{(3)}(t)-2y^{(1)}(t)+8y(t) = 2u^{(1)}(t)-u(t)$\\ + +On se ramène à une forme conforme au cours (coefficient de plus haut degré égal à 1) : +\[ y^{(3)}(t)-\frac{1}{2}y^{(1)}(t)+2y(t) = \frac{1}{2}u^{(1)}(t)-\frac{1}{4}u(t)\] + +\begin{multicols}{2} +Forme compagnon horizontal de type I : +\[ A_c = \left[\begin{array}{ccc} +0 & 1 & 0 \\ +0 & 0 & 1 \\ +-2 & 1/2 & 0 +\end{array} \right] +\quad +B_c = \left[\begin{array}{c} 0 \\ 0 \\1 \end{array} \right] \] +\[ C_c = [-1/4 \quad 1/2 \quad 0], \quad D=0\] + +Forme compagnon horizontal de type II : +\[ A_c = \left[\begin{array}{ccc} +0 & 1/2 & -2 \\ +1 & 0 & 0 \\ +0 & 1 & 0 +\end{array} \right] +\quad +B_c = \left[\begin{array}{c}1 \\ 0 \\0 \end{array} \right] \] +\[ C_c = [0 \quad 1/2 \quad -1/4] , \quad D=0\] +\end{multicols} + +Forme compagnon vertical de type I : +\[ A_o = \left[\begin{array}{ccc} +0 & 1 & 0 \\ +1/2 & 0 & 1 \\ +-2 & 0 & 0 +\end{array}\right] +\quad +B_o = \left[\begin{array}{c}-1/4 \\ 1/2 \\ 0 \end{array} \right] \] +\[ C_c = [ 1\quad 0 \quad 0] , \quad D=0\] + +\subsubsection*{Forme modale (pôles simples)} +\begin{align*} +G(p) & = \frac{p^2-1}{(p+1)(p+2)(p+3)}\\ +& = \frac{\alpha_1}{p^2-1} + \frac{\alpha_2}{p+2} + \frac{\alpha_3}{p+3} \\ +(S) & \left\{ \begin{array}{cc} +\dot{x_m} & = \left[\begin{array}{ccc} +-1 & & \\ +& -2 & \\ +& & -3 +\end{array} \right] x_m + +\left[\begin{array}{ccc} +\alpha_1 \gamma_1 \\ +\alpha_2 \gamma_2 \\ +\alpha_3 \gamma_3 +\end{array} \right] u \\ +y & = [1/\gamma_1 \quad 1/\gamma_2 \quad 1/\gamma_3]x_m+0, \quad \forall \gamma_i \neq 0 +\end{array} +\right. +\end{align*} + +\subsection{Changement de base vers une forme canonique} + +\emph{Voir polycopié} + +\subsection{Dualité observation-commande} +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x} &= Ax + Bu, \quad x(0)=x_0 \in \R^n \\ +y & = Cx+Du +\end{array} +\right. +\] + +\[G(s) = C(s1_n-A)^{-1}B+D \in \R[X]\] +$G(s)$ est scalaire, donc en transposant ($G(s)=G(s)^T, D=D^T$) : +\[G(s) = B^T(s1_n-A)^{-1}C^T+D \in \R[X]\] + +Ainsi, $\exists \tilde{x}\in \R^n$ tel que +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{\tilde{x}} &= A^T\tilde{x} + C^Tu \\ +y & = B^T\tilde{x}+Du +\end{array} +\right. +\] + +C'est la forme duale du modèle d'état (monovariable uniquement). + +\subsection{Commandabilité et observabilité pour les formes canoniques} + +Une forme canonique : +\begin{itemize} +\item de commandabilité est toujours commandable, l'observabilité est à étudier +\item d'observabilité est toujours observable, la commandabilité est à étudier\\ +\end{itemize} + + +Cas des formes modales : + +\[ +(S) : \left\{ +\begin{array}{rl} +\left[ +\begin{array}{c} +\dot{\xi_1} \\ +\vdots \\ +\dot{\xi_n} +\end{array} +\right] +& = \left[ +\begin{array}{ccc} +\lambda_1 \\ +&\ddots \\ +&& \lambda_n +\end{array} +\right] +. +\left[ +\begin{array}{c} +\xi_1 \\ +\vdots \\ +\xi_n +\end{array} +\right] ++ +\left[ +\begin{array}{c} +z_1 \\ +\vdots \\ +z_n +\end{array} +\right] +u \\ +y & = [\gamma_1 \dots \gamma_n] +\vect{z_1 \\ \vdots \\z_n} ++ D_n +\end{array} +\right. +\] +$(S)$ est commandable (resp. observable) si et seulement si tous les modes sont commandables (resp. observables) + +Si dans la matrice d'application exprimée dans la base modale, un des coefficients est nul, alors le mode correspondant, donc le système, n'est pas commandable. Il en est de même pour la matrice d'observation et l'observabilité.\\ + +Cas des formes de Jordan : (exemple) +\[ \dot{x}= \mat{\lambda_1 & 1}{& \lambda_1}{& &\lambda_1} +x + \vect{.\\.\\.\\} u, \quad u \in \R +\] +Ce n'est pas un système commandable. + +Dans un système monovariable ($y\in\R, u\in\R$), si un mode multiple est associé à au moins 2 blocs de Jordan, alors ce mode n'est pas commandable / observable. + +\section{Stabilisation par retour d'état} + +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x} &= Ax + Bu, \quad x(0)=x_0 \in \R^n \\ +y & = Cx+Du +\end{array} +\right. +\] + +Étant donné un système en boucle ouverte où $A$ peut posséder des modes / pôles instables, faiblements amortis, lents,... le but est de se donner un ensemble $\{ \lambda_1^{des}, \dots, \lambda_n^{des}\}\in\C^n$ auto-conjugué, et de chercher une loi de commande $u(t)$ permettant d'obtenir en boucle fermée un système dont les pôles / modes sont $\{ \lambda_1^{des}, \dots, \lambda_n^{des}\}$. + +\paragraph{Hypothèses : } $x(t) = \vect{x_1(t) \\ \vdots \\ x_n(t)} \in \K^n$. + +On suppose que les $x_k(t)$ sont mesurables, i.e. $x(t)$ est mesurable.\\ + +\begin{dfn} +Une loi de commande par retour d'état est une expression du type +\[u(t) = \kappa (x(t)) \text{ où } \kappa : \begin{array}{ll} +\R^n & \rightarrow \R \\ +x(t) & \mapsto u(t) = \kappa(x(t)) +\end{array} +\] +Dans le cas d'une loi de commande linéaire, la ldc par retour d'état est une expression du type : +\[ u(t) = Kx(t) \text{ où } K\in \R^{1\times n} \] +$K$ est alors appelé gain du rectour d'état. + +Une ldc linéaire par retour d'état et consigne est une expression du type : +\[ u(t) = Kx(t) + \eta e(t) \] +où $K\in \R^{1\times n}$ gain du retour d'état, $\eta \in \R$ terme de précommande et $e(t)$ signal de consigne (ou de référence). +\end{dfn} + +\subsection*{Schéma bloc :} + +\subsection*{Mise en équation :} +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x} &= Ax + Bu, \quad x(0)=x_0 \in \R^n \\ +y & = Cx+Du +\end{array} +\right. +\] + +\begin{align*} +\intertext{Avec $u=Kx+\eta e$,} +\dot{x} & = Ax + BKx + \eta Be \\ +& = (A+BK)x+\eta Be +\intertext{Posons $A_{bf} = A+BK \in \K^{n \times n}$ matrice d'évolution en bf, $B_{bf} = \eta B \in \K^{n\times 1}$ matrice d'application du signal de consigne} +\dot{x} & = A_{bf}x + B_{bf} e +\end{align*} + +\subsection*{Calcul du gain $K$ du retour d'état} + +On souhaite trouver $K \in \K^{1 \times n}$ tel que $\{ \lambda_1^{d}, \dots, \lambda_n^{d}\}$ correspondent aux valeurs propres de $A_{bf} = A+BK$.\\ + +\textbf{Hypothèse : } $(S)$ est commandable, i.e. $C(A,B)$ est inversible.\\ + +Soit $P_A(\lambda) = det(\lambda 1_n-A) = \lambda^n + a_{n-1} \lambda^{n-1} + ... + a_1 \lambda + a_0$. + +Soit $\Pi_d(p)$ le polynôme caractéristique désiré en boucle fermée. + +\begin{align*} +\Pi_d(\ld) & = \prod_{i=1}^n(\ld-\ld_i^d) \\ +& = \ld^n + \alpha_{n-1}\ld^{n-1} + ... + \alpha_1\ld + \alpha_0 , \alpha_k\in\R +\end{align*} + +Ainsi, on cherche $K=[k_0, k_1,\dots,k_n]$ tel que +\begin{align*} +P_{A_{bf}}(\ld) & = det(\ld1_n-A_{bf}) \\ +& = det(\ld 1_n - A - BK) \\ +& = \Pi_d(\ld) +\end{align*} + +Cette équation polynomiale équivaut à un système linéaire de $n$ équations à $n$ inconnues $k_0, k_1,\dots,k_n$. + +On identifie terme à terme les monômes de $\Pi_d(\ld)$ et $P_{a_{bf}}(\ld)$ pour obtenir les $n$ équations. + +\subsubsection*{Obtention de $K$ à partir de la forme canonique de commandabilité} + +$\exists M \in \K^{n\times n} $ inversible tel que : +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x_c} &= A_cx + B_cu\\ +y & = C_cx+D_cu +\end{array} +\right. +\] +où \[ A_c = \left[ \begin{array}{cccc} +0 & 1 \\ +& & \ddots \\ +& & & 1 \\ +-a_0 & \dots & & -a_{n-1} +\end{array}\right], B_c = \vect{0 \\ \vdots \\ 0 \\ 1}, x = Mx_c \] + +\begin{align*} +u(t) & = Kx(t) + \eta e(t) \\ +u(t) & = KMx_c(t) + \eta e(t) +\end{align*} + +Posons $\tilde{K} = KM = [\tilde{k_1}\dots\tilde{k_{n-1}}]$. Ainsi, +\begin{align*} +\dot{x_c} & = A_c x_c + B_c \tilde{K}x + \eta B_c e \\ +& = \tilde{A_{bf}}x_ + \eta B e \\ +\tilde{A_{bf}} & = +\left[ \begin{array}{cccc} +0 & 1 \\ +& & \ddots \\ +& & & 1 \\ +-a_0 & \dots & & -a_{n-1} +\end{array}\right] + +\vect{0 \\ \vdots \\ 0 \\ 1}.[\tilde{k_0}...\tilde{k_{n-1}}] \\ +& = \left[ \begin{array}{cccc} +0 & 1 \\ +& & \ddots \\ +& & & 1 \\ +-a_0 + \tilde{k}_0 & \dots & & -a_{n-1} +\tilde{k}_{n-1} +\end{array}\right]\\ +P_{\tilde{A_{bf}}}(\ld) & = det(\ld1_n - \tilde{A_{bf}}) \\ +& = \ld^n + (a_{n-1}-\tilde{k}_{n-1})\ld^{n-1} + \dots + (a_0-\tilde{k}_0) +\end{align*} + +Rappel : si $P_A(\ld) = det (\ld1_n -A)$, $ P_{T^{-1}AT} = det(T^{-1}) P_A(\ld) det(T)$. + +En identifiant terme à terme $ P_{\tilde{A_{bf}}}(\ld)$ avec $\Pi_d(\ld)$, on obtient +\begin{align*} +a_{n-1} - \tilde{k}_{n-1} & = \alpha_{n-1} \\ +\vdots \\ +a_{1} - \tilde{k}_{1} & = \alpha_{1} \\ +a_{0} - \tilde{k}_{0} & = \alpha_{0} \intertext{ d'où $\tilde{k}_j = a_j - \alpha_j, j=0...n-1$ et enfin, $K=\tilde{K}M^{-1}$} +\end{align*} + +\subsubsection*{Autre algorithme : formule d'Ackerman} +\[ K = [0\dots0,1].C(A,B)^{-1} \quad \avec \quad C(A,B) = [B \quad AB \quad ... \quad A^{n-1}B] \] +Remarque : il n'y a pas besoin de calculer l'inverse de $C(A,B)$ dans son intégralité, mais seulement la dernière ligne, c'est-à-dire seulement la dernière colonne de la matrice des cofacteurs. + +\[C(A,B)^{-1} = \mat{}{}{* & \dots & *} = \frac{1}{det(C(A,B))} \mat{&&*}{&&\vdots}{&&*}^T\] + +\subsection*{Calcul du terme de précommande $\eta$} +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x_c} &= (A+BK) x + B\eta u\\ +y & = (C+DK) x+ B\eta u +\end{array} +\right. +\] + +Soit $G_{bf}(p) = ((C+DK)(p1_n-A-BK)^{-1}B+D)\eta$ + +Erreur statique nulle $\Leftrightarrow G_{bf}(0)=1$ (gain statique unitaire en bf) +\[ \eta = \frac{-1}{(C+DK)(A+BK)^{-1}B-D} \] + +Remarque : en cas de boucle mal posée (quand le dénominateur est nul), on peut mettre en série du système de départ un filtre passe-bas. + +\subsection*{Poursuite de trajectoire} +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x_c} &= A_cx + B_cu\\ +y & = C_cx+D_cu +\end{array} +\right. +\] +$\exists M \in \K^{n\times n}$ inversible tel que +\[M^{-1}AM = A_c = A_c = \left[ \begin{array}{cccc} +0 & 1 \\ +& & \ddots \\ +& & & 1 \\ +-a_0 & \dots & & a_{n-1} +\end{array}\right] \qquad M^{-1}B = B_c = \vect{0 \\ \vdots \\ 0 \\ 1}\] +\[ x_c = \vect{z_1\\ \vdots \\ z_n}, \quad x = Mx_c\] + +\subsubsection*{Forme canonique de commandabilité} +\[\vect{ \dot{z_1} \\ \vdots \\ \dot{z_n}} = \left[ \begin{array}{cccc} +0 & 1 \\ +& & \ddots \\ +& & & 1 \\ +-a_0 & \dots & & a_{n-1} +\end{array}\right] . \vect{z_1\\ \vdots \\ z_n} + \vect{0 \\ \vdots \\ 0 \\ 1}u\] + +Posons $a = [-a_0 \dots a_{n-1}]$, puis $v(t) = -a^Tx_c(t)+u(t)$, on obtient la forme de Brunowsky : +\[ \vect{ \dot{z_1} \\ \vdots \\ \dot{z_n}} = \vect{z_2\\ \vdots \\ z_n \\ v} \] +On parle aussi d'une chaîne d'intégrateurs en cascade. + +\subsubsection*{Application au suivi de trajectoire} +Soit $y_d(t)$ une trajetoire désirée en boucle fermée. + +Si $y(t) = z_1(t)$ (par exemple), $\dots y^{(n)}(t) = z_1^{(n)}(t) = v$, + +soit $\epsilon(t) = y(t) - y_d(t) = z_1(t) - y_d(t),\dots,\epsilon^{(n)}(t) = v(t) - y_d^{(n)}(t)$ + +On pose \[v(t) = y_d^{(n)}(t) + k_{n-1}(y^{(n-1)}(t)-y_d^{(n-1)}(t)) + ... + k_1 (y^{(1)}(t)-y_d^{(1)}(t)) + k_0(y-y_d)\] +où les racines du polynôme caractéristique $p^n + k_{n-1}p^{n-1} + k_1p+k_0$ sont à partie réelle strictement négative, alors +\[\lim_{t\rightarrow \infty} y(t) = y_d(t) \] + +Enfin, \[u(t) = v(t) + a^Tx_c\] + +\newpage +\section{Observateur} + +\subsection{Concept} +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x} &= Ax + Bu,\quad x(0) = x_0 \in \R^n \\ +y & = Cx+Du +\end{array} +\right. +\] + +Seul $y$ est mesuré à chaque instant par un capteur.\\ + +L'observateur donne une estimation du vecteur d'état du système telle que : + +$\forall \epsilon > 0,$ arbitrairement petit, $\exists T > 0$ tel que +\[ \forall t > T, ||x(t) - \hat{x}(t) || < \epsilon \] + +But: faire la synthèse du système $(O)$ sous forme d'état, appelé observateur du système $(S)$. + +\subsection{Observateur asymptotique (extension de l'observateur de Luenberger)} + +Hypothèse : système $(S)$ observable + +\begin{dfn} +Un observateur asymptotique d'ordre $n$ est donné par le modèle d'état \[\dot{\hat{x}}=A\hat{x}+Bu+L(y- \hat{y}), \quad \hat{x}(0)=\hat{x}_0\in\R^n\] où $\hat{x}\in \R^n$ est le vecteur d'état quelconque de l'observateur et $L\in\R^{n\times 1}$ est le gain de l'observateur asymptotique. $L(y- \hat{y})$ correspond à un terme de correction, et $\epsilon_y=y-\hat{y}$ est appelé innovation. +\end{dfn} + +\textbf{\textbf{But :}} calculer $L\in\R^{n\times 1}$ tel que $\lim_{t\rightarrow\infty} ||x(t) - \hat{x}(t) || = 0$ \\ + +Soit $\epsilon_x(t) = x(t)-\hat{x}(t)\in\R^n$, on a +\[\dot{\epsilon_x}(t) = \dot{x}(t)-\dot{\hat{x}}(t) = Ax(t)+Bu(t)-(A\hat{x}(t)+Bu(t)+L(y(t)-\hat{y}(t))\] + +Or, $y(t) = Cx(t) + Du(t)$ et $\hy=C\hx(t) +Du(t)$, donc $y-\hy = C\epsilon_x$ +\[\dot{\epsilon_x}(t) = (A-LC)\epsilon_x(t)\] + +$\epsilon_x(0) = x(0) - \hx(0) = x_0 - \hx_0$ avec $x_0$ inconnu et $\hx_0$ choisi arbitrairement par l'utilisateur. + +\[\epsilon_x(t) = e^{(A-LC)t}(x_0-\hx_0)\] + +$A-LC$ : dynamique d'observation. + +Si les valeurs propres de $A-LC$ sont à partie réelle strictement négative, alors \[\limti \epsilon_x(t) = 0 \quad \text{i.e.} \quad \limti \hx(t) = x(t)\] + +Ainsi, on se donne un polynôme caractéristique désiré pour la dynamique d'observation +\[ \Pi_o(\ld) = \Pi_{i=1}^n (\ld-\ld_i^o) \] + +avec $\{\ld_j\}_{j=1..n}$ auto-conjugué (stable par conjugaison) et $\forall j, Re(\ld_j)<0$ + +\[ \Pi_o(\ld) = \ld^n + \sum_{i=0}^{n-1} \gamma_i \ld^i, \quad \gamma_k\in\R \] + +Soit $L=[l_1 \quad ... \quad l_{n-1}]^T$, le calcul de $L$ s'appuie sur la résolution du système linéaire (de type $ML=b$) issu de l'identification terme à terme des monômes de $P_{A-LC}(\ld) = det (\ld 1_n - (A-LC))$ avec ceux de $\Pi_o(\ld)$. + +Soit $T\in\R^{n\times n}$ inversible tel que $x(t) = Tx_0(t)$ conduit à la forme canonique d'observabilité, c'est-à-dire avec +\[ +\left\{ +\begin{array}{ll} +\dot{x}_0 & = \left[ \begin{array}{cccc} +-a_{n-1} & 1 \\ +& & \ddots \\ +& & & 1 \\ +-a_0 & \dots & & 0 +\end{array}\right] x_0 + \vect{b_{n-1} \\ \vdots \\ b_0}u \\ +y(t) &= [1 \quad 0 ... 0]x+Du +\end{array} +\right. +\] + +Or, $\dot{\hx} = A\hx + Bu + L(y-\hy)$ et $\hy = C\hx$. + +Posons $\hx_0$ tel quel $\hx = T\hx_0$ +\begin{align*} +& \left\{ +\begin{array}{ll} +T\dot{\hx}_0 & = AT\hx_0 + Bu + L(y-\hy) \\ +\hy & = CT\hx + Du +\end{array} +\right. \\ +\Leftrightarrow & \left\{ +\begin{array}{ll} +\dot{\hx}_0 & = A_0\hx_0 + B_0u + T^{-1}L(y-\hy) \\ +\hy & = C_0\hx + D_u +\end{array} +\right. +\end{align*} +où $ A_0 = T^{-1}AT, B_0 = T^{-1}B, C_0 = CT$ + +Posons $\tilde{L} = T^{-1}L = [\tilde{l}_{n-1} ... \tilde{l}_0]^T$ +\[ \dot{\tilde{\epsilon}}_x = (A-\tilde{L}C_0)\epsilon_x\] + +Calculons $A_0-\tilde{L}C_0 = +\left[ \begin{array}{cccc} +-a_{n-1}-\tilde{l}_{n-1} & 1 \\ +\vdots & 0 & \ddots \\ +\vdots & & & 1 \\ +-a_0-\tilde{l}_0 & & & 0 +\end{array}\right]$ + +Son polynôme caractéristique est $P_{A_0-\tilde{L}C_0}(\ld) = \ld^n + (a_{n-1}\tilde{l}_{n-1})\ld^{n-1} + ... + (a_0 + \tilde{l}_0)$ (déduit de la forme de la matrice compagnon).\\ + +En identifiant terme à terme les monômes de $\Pi_d(\ld)$ avec $P_{A-LC}(\ld)$ : $\tilde{l}_j = \gamma_j - a_j$ d'où on en déduit $L = T\tilde{L}$. + +\subsection{Correcteur par retour de sortie - Correcteur par retour d'état sur l'état reconstruit} + +\[ (S) : +\left\{ +\begin{array}{ll} +\dot{x} &= Ax + Bu,\quad x(0) = x_0 \in \R^n \\ +y & = Cx+Du +\end{array} +\right. +\] + +Loi de commande par retour d'état et consigne : +\[ u(t) = Kx(t) + \eta e(t) \] où $x$ était supposé entièrement mesurable.\\ + +En pratique on utilisera +\[ +(C) \left\{ +\begin{array}{ll} +\dot{\hx} &= Ax + Bu + L(y-\hy)\\ +\hy & = C\hx+Du +\end{array} +\right. +\] +\[u(t) = K\hx(t) + \eta e(t)\] +Correcteur dynamique par retour de sortie avec la structure observateur - retour d'état sur l'état reconstruit. + +\begin{align*} +\dot{\hx} & = A\hx + BK\hx + \eta Be + LC(x-\hx) \\ +& = (A+BK-LC)\hx + LCx + \eta BE \\ +& = (A+BK-LC)\hx + L(y-DK\hx-\eta De) + \eta Be \\ +& = (A+BK-LC-LDK)\hx + (B-LD)\eta e + Ly \\ +& = K_A\hx + K_{Be} e + Ly \\ +u & = K\hx + \eta e + 0y +\end{align*} + +\begin{prop}[Principe de séparation] +La dynamique du système $(\Sigma)$ bouclé au correcteur est donné par l'union de : +\begin{itemize} +\item la dynamique de commande (valeurs propres de $A+BK$) +\item la dynamique d'observation (valeurs propres de $A-LC$) +\end{itemize} +\end{prop} + +\begin{proof} + +\end{proof} + +\section{Discrétisation} + +\end{document} diff --git a/421-Controle_processus/Cours/pid.png b/421-Controle_processus/Cours/pid.png new file mode 100644 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AA + +\documentclass[../../Cours_M1.tex]{subfiles} +\newcommand{\nomTD}{TD1 : Systèmes échantillonés} +\renewcommand{\nomentete}{UE411 - \nomTD} + +\begin{document} + +\titre{\nomTD} + +\subsection*{Exercice 1} +\begin{itemize} +\item Montrer que $U(z) = \frac{z}{z-1}E(z)$, avec $u_k = \sum_{j=0}^{k} e_j$ et $Z(e_j) = E(z)$ + +\begin{align*} +u_{k-1} & = \sum_{j=0}^{k-1}e_j \\ +u_k & = e_k+u_{k-1} \\ +\intertext{Donc en appliquant la transformée en z et en utilisant le théorème du retard,} +U(z) & = E(z) +z^{-1}U(z) \\ +U(z) & = \frac{z}{z-1}E(z) +\end{align*} + +\item Montrer que $Z\{ke_k\}=-z\frac{d}{dz}(E(z))$ +\begin{align*} +-z\frac{d}{dz}(E(z))&= -z\frac{d}{dz}(\skzi e_kz^{-k})\\ +&=-z \skzi e_k(-k)z^{-k-1}\\ +&=\skzi ke_kz^{-k}\\ +&=Z\{ke_k\}\\ +\end{align*} +\end{itemize} + +\subsection*{Exercice 2 :} +\noindent Méthode : on effectue la transformée inverse pour obtenir le signal temporel. Puis on l'échantillonne avant de passer à sa transformée en Z, où l'on obtient une suite géométrique que l'on simplifie.\\ + +\begin{enumerate} +\item \[Y(p)= L[y(t)] = \frac{1}{p(p+a)} = \frac{\alpha}{p}+\frac{\beta}{p+a}\] + +On identifie \[\alpha=\frac{1}{a} \text{ et } \beta=\frac{-1}{a} \] + +donc par transformée inverse de Laplace, \[y(t)=\frac{1}{a}(1-e^{-at})\mathbf{1}(t)\] + +Puis on échantillonne avec $t=k.t_e$ avec $k \in\mathbb{N}$, et on a $y_k=\frac{1}{a}(1-e^{-a.k.T_e})$\\ +On a donc : +\begin{align*} +Z\{y_k\} &= Z\{\frac{1}{a}\} - Z\{\frac{1}{a}(e^{-aT_e})^k\}\\ +&= \frac{1}{a}(\frac{z}{z-1}-\frac{1}{1-e^{-aT_e}z^{-1}})\\ +Y(z) &= \frac{1}{a}(\frac{z}{z-1}-\frac{z}{z-e^{-aT_e}}) +\end{align*} + +\item On pose $Y(p)= \frac{a}{p^2+a^2} = L\{\sin(at)\}$:\\ +On a donc, \[y_k=\sin(akT_e) = \frac{e^{jakT_e}-e^{-jakT_e}}{2j}\] +Or , \[Z\{e^{jakT_e}\}=\frac{1}{1-e^{jaT_e}z^{-1}}=\frac{z}{z-e^{jaT_e}}\] +et , \[Z\{e^{-jakT_e}\}=\frac{1}{1-e^{-jaT_e}z^{-1}}=\frac{z}{z-e^{-jaT_e}}\] +d'où : +\begin{align*} +Y(z) &= \frac{z}{2j}(\frac{1}{z-e^{jaT_e}}-\frac{1}{z-e^{-jaT_e}})\\ +&=\frac{z}{2j}(\frac{e^{jaT_e}-e^{-jaT_e}}{z^2-z(2\cos(aT_e)+1)})\\ +Y(z) &=\frac{z\sin(aT_e)}{z^2-z(2\cos(aT_e)+1)} +\end{align*} + +\item On procède de même que ci-dessus avec $Y(p)= \frac{p}{p^2+a^2} = L\{\cos(at)\}$:\\ +On a donc, \[y_k=\cos(akT_e) = \frac{e^{jakT_e}+e^{-jakT_e}}{2}\] + +D'où +\begin{align*} +Y(z) &=\frac{z}{2}(2z-\frac{e^{jaT_e}-e^{-jaT_e}}{z^2-z(2\cos(aT_e)+1)}) \\ +Y(z) &=\frac{z(z-\cos(aT_e))}{z^2-z(2\cos(aT_e)+1)} +\end{align*} + +\item Ici, les échantillons sont définis par : +\[y_0=0 ,y_1=1, y_2=-1 \text{ et } \forall k > 2, y_k=0\] +On a alors : +\begin{align*} +Z\{y_k\} &= \skzi y_k.z^{-k}\\ +Y(z) &= z^{-1} - z^{-2} +\end{align*} + +\item $\forall k \in \N, y_{2k} = 0 \et y_{2k+1} = 1$. Ainsi, +\begin{align*} +Z\{y_k\} & = \skzi 1.z^{-(2k+1)} \\ +& = z^{-1} \skzi z^{-2k} \\ +& = \frac{z}{z^2-1} +\end{align*} + +\item $forall l \in \N, y_k = (-1)^{k+1} \mathbf{1}_k$. Ainsi, +\begin{align*} +Z\{y_k\} & = \skzi (-1)^{k+1}z^{-k} \\ +& = - \skzi (-1)^k(z^{-1})^k \\ +& = - \frac{1}{1+z^{-1}} \\ +& = - \frac{z}{z+1} +\end{align*} +\end{enumerate} + +\subsection*{Exercice 3 :} +\noindent Méthode : dans le cas général, on réécrit $\frac{Y(z)}{z}$ en décomposant en éléments simples puis on repasse Y(z) sous forme de série pour effectuer la transformée en Z inverse. De plus, pour les fractions rationnelles dont le dénominateur est d'ordre deux ou plus, on utilise la propriété de multiplication par une variable d'évolution : +\[TZ : \quad k^n x[k] \rightarrow (-z \frac{d}{dz})^n X(z) \] \\ + +\begin{enumerate} +\item Pour $h > 0$ et $a \in \mathbb{R^*}$ +\begin{align*} +Y(z) &= \frac{z}{z-a^h}\\ +&= \frac{1}{1-\frac{a^h}{z}}\\ +&= \skzi (a^h.z^{-1})^k\\ +Y(z) & = Z\{(a^h)^k\} +\end{align*} +\bigbreak + +\item $Y(z) = \frac{z+1}{(z-3)^2}$\\ +On pose \[\frac{Y(z)}{z}=\frac{z+1}{z(z-3)^2}=\frac{\alpha}{z}+\frac{\beta}{z-3}+\frac{\gamma}{(z-3)^2}\] +On identifie ensuite $\alpha = \frac{1}{9}$, $\gamma =\frac{4}{3}$ et pour $\beta$ on peut multiplier l'égalité par $(z-3)$ puis faire tendre $z$ vers $+\infty$ pour obtenir que $0 = \alpha + \beta$. D'où $\beta = \frac{-1}{9}$, et ainsi : +\[Y(z) = \frac{1}{9} - \frac{1}{9}\frac{z}{z-3}+\frac{4}{3}\frac{z}{(z-3)^2}\] +La transformée inverse donne alors en utilisant la propriété de multiplication : +\[y_k = \frac{1}{9}.\delta_k - \frac{1}{9}.3^k.\mathbf{1}_k+\frac{4}{3}.k.3^{k-1}.\mathbf{1}_k\] +\bigbreak + +\item $Y(z) =\frac{z+3}{z^2-3z+2}= \frac{z+3}{(z-1)(z-2)}$ +On pose \[\frac{Y(z)}{z}=\frac{\alpha}{z}+\frac{\beta}{z-1}+\frac{\gamma}{z-2}\] +On identifie ensuite $\alpha = \frac{3}{2}$, $\gamma =5$, $\beta = -4$, d'où : +\[Y(z) = \frac{3}{2} - 4\frac{z}{z-1}+5\frac{z}{z-2}\] +La transformée inverse donne alors : +\[y_k = \frac{3}{2}.\delta_k - 4.\mathbf{1}_k +5.2^k.\mathbf{1}_k\] +\end{enumerate} + +\subsection*{Exercice 5 :} +L'asservissement analogique considéré est le suivant, où $H(p) = \frac{C}{p(1+0.2p)}$ et $B_0(p) = \frac{1-e^{-T_ep}}{2}$, avec $T_e=0.2s$ et $C=5rad.s^{-1}$. +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\sbEntree{E} + +\sbComp{comp}{E} +\sbRelier[$E(z)$]{E}{comp} + +\sbBloc{cna}{CNA, BOZ}{comp} +\sbRelier[$\epsilon(z)$]{comp}{cna} + +\sbBloc[3]{sys}{H(p)}{cna} +\sbRelier[$U(p)$]{cna}{sys} + +\sbSortie[3]{S}{sys} +\sbRelier[$Y(p)$]{sys}{S} + +\sbDecaleNoeudy[4]{S}{R} +\sbBlocr[8]{can}{CAN}{R} +\sbRelieryx{sys-S}{can} +\sbRelierxy{can}{comp} + +\end{tikzpicture} +\end{figure} + +\begin{enumerate} +\item Par propriété du cours, on a : +\[T(z) = (1-z^{-1}).Z\{^*L^{-1}\{\frac{H(p)}{p}\}\}\] +On commence par poser $A(p)=\frac{H(p)}{p}=L\{a(t)\}$ et $a_n = a(n.T_e)$, donc + +\[T(z) = (1-z^{-1}).Z\{a_n\}\] +On a alors : +\[A(p) = \frac{C}{p^2(1+0.2p)} = C(\frac{\alpha}{p^2}+\frac{\beta}{p}+\frac{\gamma}{1+0.2p})\] +On trouve, $\gamma = 0.04$, $\alpha = 1$ et $\beta=-0.2$. Donc en repassant dans le domaine réel on a: +\[A(t) = C(t-0.2+0.2e^{-5t})u(t)\] +Donc en échantillonnant avec $a_n=A(nT_e)$ : +\[a_n=A(nT_e)=C(n.T_e-0.2 + 0.2e^{-5nT_e})\] +Ainsi, avec la transformée en Z on a : +\[Z\{a_n\}=C(T_e\frac{z}{(z-1)^2}-0.2\frac{z}{z-1}+0.2\frac{z}{z-\chi}) \text{ avec } \chi = e^{-5T_e}\] +d'où avec la formule initiale, on obtient : +\begin{align*} +T(z) &= \frac{z-1}{z}Z\{a_n\}\\ +&= C\frac{z-1}{z}(\frac{T_ez}{(z-1)^2}-\frac{0.2z}{z-1}+\frac{0.2z}{z-\chi})\\ +&= C(\frac{T_e}{z-1}-0.2+0.2\frac{z-1}{z-\chi})\\ +&= C.\frac{T_e(z-\chi)-0.2(z^2-(1+\chi)z+\chi)+0.2(z^2-2z+1)}{z^2-(1+\chi)z+\chi}\\ +T(z) &= C.\frac{(T_e+0.2(1+\chi)-0.4)z +(-\chi T_e-0.2\chi+0.2)}{z^2-(1+\chi)z+\chi} +\end{align*} +\medbreak +On pose :\begin{itemize} +\item $b_1 = C(T_e+0.2(1+\chi)-0.4)$ +\item $b_0 = C(0.2-\chi T_e-0.2\chi)$ +\item $a_1 = -(1+\chi)$ +\item $a_0 = \chi$ +\end{itemize} +\bigbreak +\noindent Ainsi, on a : +\[T(z) = \frac{b_1z+b_0}{z^2+a_1z+a_0}=\frac{B(z)}{A(z)}\] +\medbreak +\noindent Remarque : + +Les pôles analogiques sont : $p_1=0$ , $p_2 = -5$. + +Les pôles discrétisés sont : $p_{z_1}=1$ et $p_{z_2} = \chi$. + +(On peut le vérifier avec la relation $p_{z_i}=e^{p_iT_e}$.) + +Les zéros du temps discret dépendent de $T_e$.\\ + +\item On a donc en boucle fermée : +\begin{align*} +Y(z) = \frac{T(z)}{1+T(z)}E(z)&=\frac{B(z)}{A(z) + B(z)}E(z)\\ +&=\frac{b_1z+b_0}{z^2+(a_1+b_1)z+a_0} =F(z) E(z) +\end{align*} +Les zéros de T(z) sont aussi les zéros de F(z).\\ + +D'autre part, on cherche $G(z) = \frac{\epsilon (z)}{E(z)}$. Or : +\begin{align*} +\epsilon (z) &= E(z) - Y(z)\\ +&= (1-F(z))E(Z)\\ +&=\frac{1}{1+T(z)}E(z)\\ +&=\frac{A(z)}{A(z)+B(z)}\\ +&=G(z) E(z) +\end{align*} + +Ainsi, on a: \[G(z)=\frac{z^2 +a_1z+a_0}{z^2+(a_1+b_1)z+(a_0+b_0)}\] + +\item a -- Déterminons la relation de récurrence entrée-sortie du système en boucle fermée. On a en factorisant par $z^2$: +\[Y(z)=\frac{b_1z^{-1}+b_0z^{-2}}{1+(a_1+b_1)z^{-1}+(a_0+b_0)z^{-2}}E(z)\] +\[Y(z)(1+(a_1+b_1)z^{-1}+(a_0+b_0)z^{-2}) = E(z)(b_1z^{-1}+b_0z^{-2}) \] +d'où, avec la transformée inverse, la relation de récurrence suivante : +\[y_k+(a_1+b_1)y_{k-1}+(a_0+b_0)y_{k-2}=b_1e_{k-1}+b_0e_{k-2}\] + +\noindent b -- Déterminons la réponse à un échelon $E(z) = \frac{z}{z-1}$ : +\[Y(z) = \frac{z}{z-1}\frac{b_1z+b_0}{z^2+(a_1+b_1)z+(a_0+b_0)}\\ +\frac{Y(z)}{z}=\frac{\alpha}{z-1}+\frac{\beta}{z-p_1}+\frac{\alpha}{z-p_2}\] +d'où: +\[y_k = (\alpha+\beta p_1^k+\gamma p_2^k).\mathbf{1}_k\] +\end{enumerate} + + +\end{document} \ No newline at end of file diff --git a/421-Controle_processus/TD1/asservissementTD1.png b/421-Controle_processus/TD1/asservissementTD1.png new file mode 100644 index 0000000000000000000000000000000000000000..5a8c8ce1a6b18dfe3da5c16d8e2f472ddc577033 GIT binary patch literal 77378 zcmbT71yfx86Qywr?(Xgk5D4y0kl?`~IKkcBJy>uL?(XjH?(Xie_uYThet@mI6jcn= 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zQay-~EBI1mU53?|>O44esJMK(oJ7z`r*o~Ek(R!V`uj-Q+DiF<0XfLOE&2Q>gy?C& ztN>h!PL?n)kRdlnK)@KlrD^K}Lze(tx*$U^h!+G#U(;}QMt}tP(Csz;A-8e{{Wlf< zqap(`6cv(&NDHD7jbB=tn;Xq7a(w&{Ng)A#9xxw|B)0&cIOzYDpnG`>bFxNLBM&#^ zpUX=@|KK7A1RB%0m6XDyae zzfFxJvV26n5Lf{eWYHYLj9&bpFH(ocL__k|^FrjW?}`6vPfD@A6qNEEFcS2;lk;2M zJ2tAJ$JyPqqtaA~6T#%!ec0KO^sx*?X?2)UI##q 0$ +\item $c_2 - c_1 + c_0 > 0$ +\item $|c_0| < c_2 \Leftrightarrow -c_2 < c_0 < c_2$ +\end{itemize} + +On traduit ces conditions +\begin{itemize} +\item $c_2 + c_1 + c_0 > 0$ +\begin{align*} +& 1 + (-(1+D) + CT_eD) + (D + CT_e(1-2D)) > 0 \\ +& C ( T_e D + T_e - 2 DT_e ) > 0 \\ +& C (1-D) > 0 \\ +& C > 0 \text{ car } D =e^{-1} < 1 +\end{align*} + +\item $c_2 - c_1 + c_0 > 0$ +\begin{align*} +& 1 - (-(1+D) + CT_eD) + (D + CT_e(1-2D)) > 0 \\ +& 2 + 2D + cT_e(-3D+1) > 0 \\ +& C T_e(1-3D) > -2 -2D \\ +& C < -\frac{2+2D}{1-3D} \text{ car } 3D > 1 +\end{align*} + +\item $-c_2 < c_0 < c_2$ +\begin{eqnarray*} +-1 < & D + CT_e(1-2D) & < 1 \\ +-1 - D < & CT_e(1-2D) & < 1 - D \\ +\frac{-1-D}{T_e(1-2D)} < & C & < \frac{1-D}{T_e(1-2D)} +\end{eqnarray*} +\end{itemize} + +Le critère de Jury aboutit donc à : +\[ \boxed{ 0 < C < \min(-\frac{2+2D}{(1-3D)T_e},\frac{1-D}{(1-2D)T_e}) } \] + +\item On s'intéresse à l'asservissement analogique du même système +\[ Y(p) = \frac{H(p)}{1+H(p)} E(p) \] +L'équation caractéristique conduit à +\begin{align*} +1 + H(p) & = 0 \\ +1 + \frac{C}{p(1+\tau p)} &= 0 \\ +\tau p^2 + p + C &= 0 +\end{align*} +Le système est stable si et seulement si $C>0$.\\ +En analogique, la marge de gain est infinie (la phase n'est jamais égale à $-180^o$. En numérique, le BOZ induit un déphasage qui conduit à une limitation supplémentaire pour $C$ en terme de stabilité. + + +\item Pour approcher le comportement basse fréquence de la chaîne directe de l'asservissement de la figure 1, il faut tenir compte du BOZ. Il faut approximer l'expression de $B_0(p)$. + +\begin{itemize} +\item Avec $\omega << \frac{1}{T_e}$, $e^{-T_ep} \approx 1 - T_ep$ et +\begin{align*} +B_0(p) & = T_e \\ +\tilde{H}(p) & = T_e H(p) = \frac{CT_e}{p(1+\tau p)} +\end{align*} +On revient à la même condition $ C > 0 $. + +\item Suggestion : faire le développement à l'ordre 2 ? + +Avec $e^{-T_e p} \approx 1 - T_ep + \frac{(T_ep)^2}{2}$, +\begin{align*} +B_0(p) & = \frac{T_ep-T_e^2p^2 / 2 }{p}\\ +& = T_e - \frac{T_e^2}{2}p \text{ : non causal} \\ +\tilde{H}(p) & = T_e(1-\frac{T_e}{2}p)\frac{C}{p(1+\tau p)} +\end{align*} +L'équation caractéristique $1+\tilde{H}(p) = 0$ mène à +\begin{align*} +\tau p^2 + p + CT_e(1-\frac{T_e}{2}p) & = 0 \\ +\tau p^2 + (1-C\frac{T_e^2}{2})p + CT_e & = 0 +\end{align*} +L'application du critère de Routh mène à : $0 < C < \frac{2}{T_e^2}$ + +\item Approximation de Padé ($\omega << \frac{1}{2T_e}$) : +\[ e^{-T_ep} = \frac{e^{-\frac{T_e}{2}p}}{e^{\frac{T_e}{2}p}} \text{ et } e^{-\frac{T_e}{2}p} \approx 1 - +\frac{T_e}{2}p \Longrightarrow e^{-T_ep} \approx \frac{1 - \frac{T_e}{2}p}{1 + \frac{T_e}{2}p} \text{ : causal !} \] + +\begin{align*} +B_0(p) & \approx \frac{1-\frac{1 - \frac{T_e}{2}p}{1 + \frac{T_e}{2}p}}{p} \\ +& = \frac{T_e}{1+\frac{T_e}{2}p} \text{ : causal } \\ +\tilde{H}(p) & = \frac{T_e}{1+\frac{T_e}{2}p} . \frac{C}{p(1+\tau p)} +\end{align*} + +L'équation caractéristique $1+\tilde{H}(p)=0$ mène à +\begin{align*} +CT_e + p (1+\frac{T_e}{2}p)(1+\tau p) & = 0 \\ +\tau\frac{T_e}{2}p^3 + (\tau+\frac{T_e}{2})p^2 + p + CT_e &= 0 +\end{align*} + +\begin{figure}[h!] +\centering +\begin{tabular}{|c|c|c|} +\hline +$p^3$ & $\tau \frac{T_e}{2}$ & 1 \\ +\hline +$p^2$ & $\tau + \frac{T_e}{2} $ & $CT_e$ \\ +\hline +$p$ & $1- \frac{CT_e \tau}{2(\tau + \frac{T_e}{2})}$ & 0 \\ +\hline +$p^0$ & $CT_e$ & \\ +\hline +\end{tabular} +\end{figure} + +La condition est donc $C < \frac{2(\tau + T_e/2)}{T_e \tau} = \frac{3}{\tau}$ +\end{itemize} + +\item La discrétisation d'un asservissement en temps continu dégrade la stabilité. +\end{enumerate} + +\end{document} \ No newline at end of file diff --git a/421-Controle_processus/TD3/TD3.pdf b/421-Controle_processus/TD3/TD3.pdf new file mode 100644 index 0000000000000000000000000000000000000000..26b4a6919d427e4d8be60336bfc1b69f38de1ae4 GIT binary patch literal 119524 zcma&NLzE~=uxMF4W!tuG+qP}nwr$(CZQHi(s=nQ;-{}3p&rxK|A}6tS#7+``Q?#yNOpXBd^`1Out@#*-uJ$zmsomD4tUr*>4xuYkVv0=_an$!K*ONte=7K4V?DS#Eqvf-CCiX($bH`V*P)Ao% z07&xBIjPg4HRmK^ul6Zt!?kN@)YhJa>`BU7dyEX#4y$)(tu1N>nQz3qskAK2L{lNh zbZ?bnoRQyN=^(>}l4k#glLBaWmk!M%&1Cn4v!dTJb+db~`Yi0kbf;X_(=-2KX7%gs z?hv!{j?)`DMvEuZ1XV8DAX*;i>XLKpOPl6m{eYrfj~xkQ1RZs zIZ2a7H#yfzHf^ox_%O9<#k42aXWZt>rw+&p0c=jGYf^^&7(M&Z0<9-FG>ZIry>jdm z`)E_ShSg1CicefK2Uq{{*If3icf;Iwo6S*{*nb6z(YE~7xUZupmj+IQzDqF_PxXbC zMeF3>Q|ea9m)~FO29dW6>ji7}-gW(UUmlH{%ckA|P1Ei^7{_{dUlpfU@y1KzE(ukTh+drO~=Z#T0mBNczuEbpbxa>&e0_XV0tjCE z0~r!(t(YI6xkCfl&iA_6`8M)rR{DUYN?H;Kmcizrc}Ni2Os!mfZ;JOH#m%Nm1lJe! 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\frac{1}{p(1 + \tau p)} \avec \tau = 0,2\] +La période d'échantillonnage est $T_e = 0,2s$ et l'on souhaite régler le correcteur PI pour satisfaire le cahier des charges suivants en boucle fermée : +\begin{itemize} +\item marge de phase de 45 degrés +\item erreur en vitesse $\epsilon_v$ nulle +\end{itemize} + +\begin{enumerate} +\item Synthèse d'un correcteur à temps continu + +\begin{enumerate} +\item On considère le BOZ $B_0(p)=\frac{1-e^{-T_ep}}{p}$ que l'on cherche a approché par un retard pur équivalent : +\begin{align*} +B_0(p) &= \frac{1-e^{-T_ep}}{p}\\ +&= e^{-\frac{T_ep}{2}}\frac{e^{\frac{T_ep}{2}}-e^{-\frac{T_ep}{2}}}{p} +\intertext{Or, le développement limité à l'odre 1 $e^{-T_ep} = 1-T_ep$ donne :} +\tilde{B}_0(p) &= e^{-\frac{T_ep}{2}} \frac{1+\frac{T_ep}{2}- (1-\frac{T_ep}{2})}{p}\\ +& \boxed{\tilde{B}_0(p) = T_e e^{\frac{-T_ep}{2}}} +\end{align*} +\bigbreak + +\item On se propose de régler le correcteur PI en utilisant le tracé de la fonction de transfert en boucle ouverte du système.\\ +Le correcteur a donc la forme : \[C(p) = K_p(1+\frac{1}{T_ip})\] et on pose \[L(p) = \tilde{B}_0(p)H(p) = \frac{T_e e^{\frac{-T_ep}{2}}}{p(1 + \tau p)}\] + +On s'intéresse dans un premier temps à l'erreur en vitesse. On prend donc pour entrée $e(t) = t$ d'où $E(p) = \frac{1}{p^2}$\\ + +On a donc comme écart $\epsilon(p) = \frac{E(p)}{1+L(p)C(p)}$, et on va appliquer le théorème de la valeur finale : +\begin{align*} +\epsilon_v &= \lim_{t \rightarrow \infty}\epsilon(t)\\ +&= \lim_{p \rightarrow 0} p \epsilon(p)\\ +&= \lim_{p \rightarrow 0} p \frac{1}{p^2} \frac{1}{1+L(p)C(p)}\\ +&= \lim_{p \rightarrow 0} \frac{T_ip(1+\tau p)}{T_ip^2(1+\tau p)+T_ee^{\frac{-T_ep}{2}}K_p(1+T_ip)} +&= 0 +\end{align*} +L'erreur statique de vitesse est bien nulle avec ce modèle du retard pur pour le BOZ.\\ + +Intéressons-nous maintenant au réglage du correcteur, on relève le gain lorsque la phase est de -135 degrés et on cherche à annuler ce gain pour imposer le marge de phase à 45 degrés. On relève un gain de -35dB, il faut donc relever le gain de 35dB : +\[20log(K_p) = 25 \Rightarrow K_p = 10^{\frac{25}{20}} = 18\] + +De plus, pour régler $T_i$, on fait en sorte que la phase du correcteur n'influence pas trop celle du système dans la bande passante, donc que la phase du correcteur seul soit à zéro quand celle du système est proche du point critique -1. Il est une bonne approximation de prendre $T_i$ de sorte que $\frac{1}{T_i}$ soit une décade plus tôt que la pulsation de coupure du système ou d'annulation du gain (choix effectué ici). Donc : +\[\frac{1}{T_i} = \frac{\omega_{0dB}}{10} \Rightarrow T_i = \frac{10}{3} = 3.3s\] + +\item On utilise l'approximation $z=e^{pT_e} \approx 1+pT_e$ pour écrire la fonction de transfert en $z$ du correcteur échantillonné, on a donc $p = \frac{z-1}{T_e}$, d'où :\\ +\begin{align*} +C(p)|_{p = \frac{z-1}{T_e}} &= C(z)\\ +&= K_p(1+ \frac{1}{\frac{T_i}{T_e}(z-1)})\\ +&=K_p\frac{z+\frac{T_e}{T_i}-1}{z-1} +\end{align*} + +La fonction de transfert en $z$ du système + BOZ est : +\begin{align*} +T(z) & = (1-z^{-1})Z\{^*L^{-1}\{\frac{H(p)}{p}\}\} \\ +& = \frac{b_1z+b_0}{z^2+a_1z+a_0} \\ +\avec b_0 & = (1-D)\tau - T_eD \\ +b_1 & = T_e + (D-1)\tau \\ +a_0 & = D \\ +a_1 & = -D-1 \\ +D & = e^{-T_e/\tau} +\end{align*} + +\item Déterminons si le système échantillonné est stable si asservi de la sorte (correcteur numérique) :\\ +On peut appliquer le critère de Routh à $1+C(z)T(z)$, avec $T(z)$ la fonction de transfert de l'asservissement échantillonné, CF TD précédent. +\end{enumerate} + +\item Synthèse "pseudo-continue" + +\begin{enumerate} +\item Déterminons la fonction de transfert en $w$ du système échantillonné : +\begin{align*} +T(z)|_{z=\frac{1+w}{1-w}} = \tilde{T}(w) &= \frac{b_1\frac{1+w}{1-w}+b_0}{(\frac{1+w}{1-w})^2 + a_1 \frac{1+w}{1-w} + a_0}\\ +&= \frac{b_1(1-w^2)+b_0(1-2w + w^2)}{(w^2+w+1)+a_1(1-w^2)+a_0(1-2w+w^2}\\ +\end{align*} +\[\boxed{\tilde{T}(w)= \frac{(b_0-b_1)w^2-2b_0w + b_0+b_1}{(1-a_1+a_0)w^2 + 2(1-a_0)w + 1+a_1+a_0}}\] + +\noindent Comment se traduit le cahier des charges pour le système transformé ?\\ +\[ z= \frac{1+w}{1-w} \Leftrightarrow w = \frac{z-1}{z+1} \text{ avec } z= e^{T_ep} \] +La réponse fréquentielle est valable pour $p = j\omega$ avec $\omega \in [0,\frac{\pi}{T_e}]$, donc pour : +\[w = \frac{e^{j\omega T_e}-1}{e^{j\omega T_e}+1} = \frac{2j\sin(\omega \frac{T_e}{2})}{2\cos(\omega \frac{T_e}{2})} = j\tan(\frac{\omega T_e}{2})\] +Ainsi, quand $\omega$ varie de 0 à $\frac{\pi}{T_e}$ alors $w$ varie de 0 à l'infini sur l'axe des imaginaires.\\ +On pose donc $w = j\tilde{\omega}$ et $\tilde{\omega}\in[0;+\infty[$ \\ + +En quoi on répond à la question??? + +\item On va régler ici le correcteur PI en la variable $w$ : +On a $C(w) = K_p(1+\frac{1}{T_iw}$ et $\epsilon(w) = \frac{1}{1+C(w)\tilde{T}(w)}E(w)$\\ +On applique une entrée en rampe $t \rightarrow nT_e \rightarrow \frac{T_ez}{(z-1)^2}$, on a donc : +\begin{align*} +E(w) = E(z)|_{z=\frac{1+w}{1-w}} &= \frac{T_e(1+w)}{(1-w)(\frac{1+w-1+w}{1-w})^2}\\ +&=\frac{T_e(1-{w}^2)}{4{w}^2} +\end{align*} + +Intéressons nous maintenant à l'écart statique en vitesse avec el théorème de la valeur finale: +\begin{align*} +\epsilon_v &= \lim_{z \rightarrow 1} (z-1) E(z)\\ +z \rightarrow 1 &\Leftrightarrow w \rightarrow 0 \text{ car, $z-1 = \frac{2w}{1-w}$}\\ +\epsilon_v &= \lim_{w \rightarrow 0} \frac{2w}{1-w}\epsilon(w)\\ +&= \lim_{w \rightarrow 0} \frac{2w}{1-w}\frac{1}{1+K_p(1+\frac{1}{T_iw}\frac{\beta_2{w}^2+\beta_1 w + \beta_0}{\alpha_2{w}^2+\alpha_1 w + \alpha_0}\frac{T_e(1-{w}^2)}{4{w}^2})}\\ +&= ...\\ +&= 0 +\end{align*} + +Caractérisons maintenant le correcteur en fonction du cahier des charges. Comme précedemment on relève le gain lorsque la phase est à -135 degrés et on a : $K_p = 10^{\frac{10}{20}} = 3.3$ et le choix de $T_i$ est le même fait comme précedemment donc : $T_i = \frac{10}{\tilde{\omega}_{0dB}} = 33.3s$\\ + +\item La transformée en z ainsi, obtenue est : +\begin{align*} +C(z) &= C(w)|_{w = \frac{z-1}{z+1}}\\ +&= K_p(1+\frac{z+1}{T_i(z-1)}\\ +&= \frac{C_p}{T_i}\frac{T_iz+1-T_i}{z-1} +\end{align*} +\end{enumerate} +\end{enumerate} + +\end{document} diff --git a/421-Controle_processus/TD4/TD4.pdf b/421-Controle_processus/TD4/TD4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b7fd29899f37d52ce6d137b0b7caf68768b5b9a8 GIT binary patch literal 139769 zcma&MQ+Fm>w1yemPR0I`if!ArZQHhO+qP|+729_2?$LMWT%38ee!v`aJuk@Qg+*x? zXjx#$=9fp-Vb}@j3GECm3Awpp=%h_-&792%8QECq3ID%=p%b;Rb~bS&q!YC^a5fP( zF|so@f#KzaadLJvF|dJg-@Mk=a>8jt_g$-7Z%a%<0!O&XmyAsOD$g7Tq4*D8o5DPAJH|h18~elNhA>3+tcoK)LOhqi8iHS&ZS}ceI-p$H^3;j zdS_`!`QJ-RiA1AsJdk(gerL>JWm+cI{n1JQMl#JreYFYR>La`?a@g>waAZ5)m;QjU@$OSvEa)MP0HncMaV7 zjsqH0O@3)rd#At-g)Jj)?V75h`gG?X7@51#l(12QVZrxX_z%R6Tewp+IjMG}gvow% z6wliXhb1+vMVi3v4ei?u)24opx}HsQKKJU)$IL!`dOux)etV~GPqJqaejcwYxeb2@ z{J!Ei#y6~B=+m~RAACBtW(aoo-rX)~Hw8ELeQ0!T2jKQcJs*A?Pc?db`?lRZoNeTj5<#0FT?J4>+L-QjreLcDUq~bCOs2OyM=KF#3Y8`k zCO(!HI#|U@{v*KucUgVe9DCi?h(ej(q)C5aY&m3LxiYDd{)m^qZb z=!3L^{l?fxB2zs6SalPr=$~;HfA~fFx0R$tmcU4tCA8DQ6?-&?GY+S83!;@n7w@&m z=if#?=!`H=+fU;y^={XMs5BvHDqeOPTaN`H7M@OJoWC4;OnBiYL zoeYReG~jYJD!i7@9GXIiwz$&rOWw)|Sy(rRLQ{?Uv#sPxMI{{aX1gAJ5KJzd<7)Ly zo1(M66-<*t%jO8GWF96Taw_DB_6zVWN4kmFfcr640|ZKNtAas9_cAKbAn!){v$8~_e;vl9JI;Wx=vL>h|3akVd`>BeIPjX** z8B*;0;)8ACdzWC6VQ6i)cgvPOay1og6mn0K152bGh;oAG-8!YiTv>IO^w}If;|-OW zR&ivbd(37KNf`-N5$#BF&@JUe1#pJ0`i)U#KK=);!35<@#hIwcYt46>A1Z6r4y~92+P@NE7|X0EKzLCi*b%`19K+Vuu=#<=E6f$qc{7<`U zkzTV9i1}i-yRCUdmcGu_u~bJysxI-Ka+D*WMu&nl+E?d1Kv~(y*rJLBF?^Z+PlZ5v z{7yt8JJ1DfErF?9$axMF3`PU-$&1QCXgGB`J%UhHPNtHbq;*j#fPgl#&`V;-jb}bZ z5YQ}IbY+K>zYN5GPwQ`(AHi@X?0+x>R5A-AKX^B2=w3FmTrY@@tSlQSpKRExFGiTm zjaP6`pQC}dO9$SsYWP4lx@`xA5~~INszuHK?1!Rz=P=KtD2JwN(-Q}y@Zc7n2~`M2 z1|w7rF51?8p(W7RE&t4-wnHPaPvzk=<2@ z*$WU#KdTAFAx(a6ho9AU0VYukVX#2&vQ09Pr4p)Jfag7MSawLR;I&JGaz=h7rP?L0 z?YXRf;>F$p^}^M_a;fvK9z`}V!3(-WTXBCBX|^uph!O6|))1Mmpa#2O%b{;h=l}8q zDdvR+kC>UN_k`kG^LTosPB1<;H$@`4^5KK;nRSD*+S%m>de?N1^e$ zqaRdLj@7x!Q$#nkMV;TdOWB5Z-iTJ(ySu2;R)RbH)Evt$8WORC#t>N9ob}o?3@69( zC{KbX-IY}K(vb$kTiNZ9oyq68UH!Az&uljw4?%$MW3MY9!}5{;DMrw!Kf}7RJ+=(1 z7LbZH`-CwHCZsXhd(vBqrF##DRyw3|qLZBw@b}59;_eaRrjO?^ntP0-5qXU;7a=eh z9}IdX@?ruC9Tx(cCNfl33mgZOhmh34i2z8%jq=aK^JC8IC-(?@6vrUt62j@E)dlPE zR5F(TgHor*mNHwM0?&MDF@H2e#vu)xW4;`vhh1JGaV*J#KOE-b%du$f7FzN~-;3b% z6y@b-{a^syv$ZWpPfy)w(~kAa_l=IWE7hfJR@dSBwr$?nbw7dH*47H=ZEgO)l_&#U zO_?M=@d(^uiNlhx7yRsTKzAIZ4JT{NX|bBb7{wq1bnX?DNUwvX@b#bS<^Av_KxngF zn7QTjUH++r|HrYz0&6=~LahAcI%UEfV*njO9FZfqN2Im#x@b7>CVFk6YRU4EDog#? 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RST} +On considère la fonction de transfert: \[G(p) = \frac{1}{p(1+\tau p)}\] + +Rappel : (TD précédent) +\begin{align*} +G(z) & = \frac{B(z)}{A(z)} \\ +& = \frac{b_1z+b_0}{(z-1)(z-D)} \avec D=e^{-T_e/2}\\ +& =\frac{b_1z+b_0}{z^2+a_1z+a_0} \\ +n & = deg(A) = 2 \\ +m & = deg(B) = 1 \\ +\end{align*} + +\begin{enumerate} + +\item Il y a plusieurs façon de représenter le correcteur RST : +\begin{itemize} +\item $\nu(z)=-\frac{S(z)}{R(z)}Y(z) + \frac{T(z)}{R(z)}E(z)$ + +Les conditions de causalité sont alors : $\tau \leq p$ et $\sigma \leq p$.\\ + +\item $\nu(z)=\frac{S}{R}[-Y(z) + \frac{T}{S}E(z)] $ + +Causalité : $\tau \leq p$ et $\sigma \leq p$.\\ + +\item $\nu(z)=\frac{T}{R}[-\frac{S}{T}Y(z) + E(z)]$ + +Causalité : $\tau \leq p$ et $\sigma \leq p$.\\ +\end{itemize} + + +\item On a : $Y(p)= G(p)[\nu(p) + P(p)]$, $P(p) = \frac{P_0}{p}$ et $\nu(p) = B_0(p)\nu^*(p)$\\ + +\begin{align*} +Y(p) &= G(p)[\nu(p) + P(p)]\\ +&= G(p)[B_0(p)\nu^*(p)+\frac{P_0}{p}]\\ +&= \frac{1-e^{-T_ep}}{p}G(p)\nu^*(p) + G(p)\frac{P_0}{p}\\ +\intertext{On pose $A(p) = \frac{G(p)}{p}$, et on a après transformée inverse de Laplace :} +y(t) &= (a*\nu^*)(t)-(a*\nu^*)(t-T_e)+a(t)P_0\\ +\intertext{après discrétisation on obtient : } +y_k & = a_k*\nu_k-a_{k-1}*\nu_{k-1} + a_kP_0\\ +\intertext{après transformée en $z$ on a alors :} +Y(z) &= A(z)\nu(z)-z^{-1}A(z)\nu(z)+A(z)P_0\\ +&=(1-z^{-1})A(z)\nu(z)+A(z)P_0\\ +&= G(z)\nu(z)+(1-z^{-1})A(z)(1-z^{-1})^{-1}P_0\\ +&= G(z)\nu(z)+G(z)P_0\frac{z}{z-1}\\ +Y(z)&= G(z)(\nu(z)+p(z)) +\end{align*} +\bigbreak + +\item Calcul de R et S. + +On considère l'exigence 1 du cahier des charges :\\ +$p_{1,2}=\omega_0 (-\xi \pm j\sqrt{1-\xi^2})$, avec $\omega_0 = 1.5$ et $\xi=0.7$ + +$p_3 = -3\omega_0 \xi$\\ + +Comme $z_i = e^{T_ep_i}$, avec $\Omega = \omega_0 \sqrt{1-\xi^2}$, + +$z_1 = e^{-T_e\omega_0\xi}(cos(\Omega T_e) +j sin(\Omega T_e))$ + +$z_2 = \overline{z_1}$ + +$z_3 = e^{-3\omega_0 \xi T_e}$\\ + +En boucle fermée, l'asservissement considéré donne en posant $G(z) = \frac{B(z)}{A(z)}$, +\[Y(z) = \frac{BT}{AR+BS}E(z) + \frac{BR}{AR+BS}p(z)\] +On pose $H_d(z) = \frac{B_d(z)}{\Pi_d(z)} = \frac{BT}{AR+BS}$, avec $deg(\Pi_d)=q$ et $deg(B_d)=\mu$. Les pôles (continus) imposés par le cahier des charges conduisent à la forme suivante pour le polynôme caractéristique : +\begin{align*} +\Pi_d(z) &= (z-z_1)(z-z_2)(z-z_3)\\ +&= z^3+C_2z^2+C_1z+C_0\\ +\intertext{avec} +C_0 &= z_1z_2z_1 = -e^{-5\omega_0 \xi T_e}\\ +C_1 &= z_1z_2 + z_2z_3 + z_3z_1 = (e^{-2 \xi \omega_0 T_e}+2e^{-4\xi \omega_0 T_e}cos(\Omega T_e))\\ +C_2 &= -z_1-z_2-z_3 = -(e^{-3\xi \omega_0 T_e}+2e^{-\xi \omega_0 T_e}cos(\Omega T_e))\\ +\end{align*} + + + +On s'intéresse à l'exigence 2 du cahier des charges : +Pour $E(z) = 0$ i.e $e_k = 0$, et $P(z) = P_0 \frac{z}{z-1}$, +\[\lim_{\rightarrow\infty} y_k =0\] + +Ainsi, avec le théorème de la valeur finale on a : +\begin{align*} +\lim_{z\rightarrow 1} \frac{z-1}{z}Y(z) = \lim_{z\rightarrow 1} \frac{BR}{AR+BS}P_0 =0 +\Leftrightarrow & \lim_{z\rightarrow 1} B(z)R(z) = 0\\ +\Leftrightarrow & R(z) = (z-1)^l \tilde{R}(z)\text{\indent avec, } l\geq 1\\ +\end{align*} + +Par simplicité, on prend donc $l=1$. + +\[\frac{B(z)T(z)}{A(z)R(z)+B(z)S(z)} = \frac{B_d(z)}{\Pi_d(z)}\] +donc $B(z)T(z) = B_d(z)$ et $A(z)(z-1)\tilde{R}(z)+B(z)S(z)=\Pi_d(z)$\\ +donc avec $\tilde{A}(z) = (z-1)A(z)$, \[\boxed{\tilde{A}(z)\tilde{R}(z) + B(z)S(z) = \Pi_d(z)}\] + + +Soit $\tilde{n} = deg(\tilde{A}) = n+1$ et $\tilde{\rho} = deg(\tilde{R})$\\ + +1) Égalité des degrés : $deg(\tilde{A}\tilde{R})=deg(AR)S)>XJ*Zsndsq^x17D-vdgpg5x#$0Q&Rfs4O-zU zT+wG*X@P43?oXU(Ra9uDi~^q72}r41dt1A>+tW(jx3;(Q61Walct8t-(b~D${;l}; z{=aPSk9xFHx{kJ90_Fm8f9nfK|Lqq%{qyenKku&p^X|q!?<{Gh)ZKmDyanX`UGF#k zUGMij-EDnr>^ud;?pS;JSX`62rs-*C_xB94@b`4r@rkFkXTaaf>sO_(Uc3D7I+FQ! z9o@Bav-froko`;2zj0}z|HR48#+z2k(C3NwKls*m1^K0wx?|^P@8B(PT~SV+R?67X z7HoK3`kKsNVqM%l4WC)tfMayq&eze#?t#`VTB%!(-d^|ZJk{M@pSio)f$S*IN@+T} zc!PACgB8_{ZrjbHFJoCIA9n{i8h05jg#iYXA@d&j0|)qWaIJS*QN3^qfK#NMmGNIX~?1eOD*8P70X zlvX>-Y+y}&$%Ey3aQugJg11VVSReEegl;_X456WAV+Yv~77-P@d__i9PF_J#N&U8l zrq&&89YdpskBm)B&1`J#p4vM&I(d2f`1<(=Jb(Er^mW*q@Q8%Ow@Jw<@7||o=j7&n 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+\item $i = i_3 + i_4$ +\item $i_1 = i_q + i_3$ +\item $i_4 = i_q + i_2$ +\end{itemize} + +Lois des mailles :\\ +\begin{itemize} +\item $R_1i_1 - R_2i_2 - v = 0$ +\item $R_3i_3 + v -R_4i_4 = 0$ +\end{itemize} + +\begin{align*} +R_1i_q + R_1i_3 - R_2i_2 - v = 0\\ +R_3i_3 + v -R_4i_2- R_4i_q = 0 +\intertext{ou encore :} +-R_1C\frac{dv}{dt} - v + R_1i_3 - R_2i_2 = 0\\ +R_4C\frac{dv}{dt} + v R_3i_3 -R_4i_2 = 0 +\end{align*} + +Posons $\Delta = R_1R_4-R_2R_3$ +On a donc : +\begin{align*} +i_3 &= \frac{1}{\Delta}(-(r_1+r_2)R_4C\dot{v} - (R_2+R_4)v)\\ +i_2 &= \frac{1}{\Delta}(-(R_3+R_4)R_1C\dot{v}-(R_1+R_3)v) +\intertext{on a donc :} +u &= L\frac{di}{dt}+R_1i_1+R_3i_3\\ +&= L\frac{di}{dt} - (R_1 + R_3)i_3 +R_1i_q\\ +& \vdots\\ +u &= L\frac{di}{dt}-\frac{(R_1R_2R_3+R_1R_2R_4+R_1R_3R_4+R_2R_3R_4)C}{\Delta}\dot{v} - \frac{(R_1+R_3)(R_2+R_4)}{\Delta}v +\intertext{Or, $i = i_1 + i_2$ donc $i = i_q + i_3 + i_2$, donc :} +i &= -C\dot{v} - \frac{R_1+R_2+R_3+R_4}{\Delta}v - \frac{(R_1R_3+2R_1R_4+R_2R_4)C}{\delta}\dot{v}\\ +&\vdots\\ +i &= -\frac{(R_1+R_2)(R_3+R_4)}{\Delta}\dot{v} - \frac{R_1+R_2+R_3+R_4}{\Delta}v +\end{align*} + +On pose : +\begin{itemize} +\item $\alpha_1 = R_1R_2R_3+R_1R_2R_4+R_1R_3R_4+R_2R_3R_4$ +\item $ \alpha_2 = (R_1+R_2)(R_3+R_4)$ +\item $\beta_1 = (R_1+R_3)(R_2+R_4)$ +\item $\beta_2 = R_1+R_2+R_3+R_4$ +\end{itemize} +\bigbreak +On a alors : +\begin{align*} +u = L\frac{di}{dt} - \frac{\alpha_1C}{\Delta} \dot{v} - \frac{\beta_1}{\Delta}v\\ +i = -\frac{\alpha_2C}{\Delta}\dot{v} - \frac{\beta_2}{\Delta}v +\end{align*} + +Posons $x_1 = i$ et $x_2 = v$ : +\begin{align*} +u = L\dot{x_1} - \frac{\alpha_1C}{\Delta} \dot{x_2} - \frac{\beta_1}{\Delta}x_2\\ +x_1 = -\frac{\alpha_2C}{\Delta}\dot{x_2} - \frac{\beta_2}{\Delta}x_2 +\end{align*} + + + +On a donc le système matriciel suivant : +\begin{align*} +\begin{pmatrix} +L & -\frac{\alpha_1 L}{\Delta} \\ +0 & -\frac{\alpha_2 L}{\Delta} +\end{pmatrix}.\begin{pmatrix} +\dot{x}_1 \\ +\dot{x}_2 +\end{pmatrix} &= \begin{pmatrix} +0 & \frac{\beta_1}{\Delta}\\ +1 & \frac{\beta_2}{\Delta} +\end{pmatrix}.\begin{pmatrix} +x_1\\ +x_2 +\end{pmatrix}+\begin{pmatrix} +1\\ +0 +\end{pmatrix}u +\intertext{On en déduit alors pour $\frac{\alpha_2CL}{\Delta} \neq 0$ l'équation d'état:} +\begin{pmatrix} +\dot{x}_1\\ +\dot{x}_2 +\end{pmatrix} &= \begin{pmatrix} +-\frac{\alpha_1}{\alpha_2L} & \frac{\beta_1}{\Delta}- \frac{\alpha_1\beta_2}{\alpha_2L\Delta}\\ +-\frac{\Delta}{\alpha_2C} & -\frac{\beta_2}{\alpha_2} +\end{pmatrix}.\begin{pmatrix} +x_1\\ +x_2 +\end{pmatrix} + \begin{pmatrix} +\frac{1}{L}\\ +0 +\end{pmatrix}u\\ +\intertext{Sans oublier l'équation d'obeservation :} +y&= i = x_1 = \begin{pmatrix} +1 & 0 +\end{pmatrix}.\begin{pmatrix} +x_1\\ +x_2 +\end{pmatrix} + 0*u +\end{align*} +\bigbreak + +\noindent 2- Théorème de caractérisation de la commandabilité :\\ +On considère le système suivant : +\[(S)= \left \{ + \begin{matrix} + \dot{x} = Ax + Bu & x(0) = x_0 \in \mathbb{R}^n\\ + y = Cx + Du & + \end{matrix} + \right. +\] + +\noindent (S) est commandable $\Leftrightarrow$ rang[C(A,B)] = n (matrice de rang plein)\\ +Où C(A,B) = $\begin{pmatrix} +A^0B & AB & A^2B & ...& A^{n-1}B +\end{pmatrix}\in \mathbb{R}^{n*n}$\\ + +Remarque : on est en monovariable $\Leftrightarrow$ $det (C(A,B)) \neq 0$\\ + +On calcul donc : +\begin{align*} +A &= \begin{pmatrix} +-\frac{\alpha_1}{\alpha_2L} & \frac{\beta_1}{\Delta}- \frac{\alpha_1\beta_2}{\alpha_2L\Delta}\\ +-\frac{\Delta}{\alpha_2C} & -\frac{\beta_2}{\alpha_2} +\end{pmatrix}\\ +A^0B &= \begin{pmatrix} +\frac{1}{L}\\ +0 +\end{pmatrix}\\ +AB &= \begin{pmatrix} +-\frac{\alpha_1}{\alpha_2L^2}\\ +-\frac{\Delta}{\alpha_2LC} +\end{pmatrix} +\intertext{On a donc :} +C(A,B) &= \begin{pmatrix} +\frac{1}{L} & -\frac{\alpha_1}{\alpha_2L^2}\\ +0 & -\frac{\Delta}{\alpha_2LC} +\end{pmatrix} +\intertext{On calcul alors le determinant :} +det(C(A,B)) = \frac{-\Delta}{\alpha_2L^2C} \neq 0 &\Leftrightarrow \Delta \neq 0 \\ &\Leftrightarrow R_1R_4 - R_2R_3 \neq 0 +\end{align*} +\bigbreak +Remarque : si $R_1R_4 = R_2R_3$ le pont est équilibré et $v(t) = 0$ donc le système est non commandable. + + +\section*{Exercice 2 :} +\noindent 1-a) \begin{align*} +P_a(\lambda) &= det\begin{pmatrix} +-1-\lambda & 1 & 0\\ +-1 & -3-\lambda & 0\\ +1 & 1 & -2-\lambda +\end{pmatrix}\\ +&= (-1-\lambda)(-3-\lambda)(-2-\lambda) + (-2-\lambda)\\ +&= (-2-\lambda)(+3+\lambda +3\lambda + \lambda^2+1)\\ +&= (-2-\lambda)^3 +\end{align*} + +\noindent 1-b) $\lambda_0 = -2$ vecteur propre triple\\ + +\noindent 1-c) Cherchons les vecteurs propres vérifiant $AX = \lambda X$: +\begin{align*} +\begin{matrix} +-x_1 + x_2 &= -2x_1\\ +-x1 -3x_2 &= -2x_2\\ +x_1+x_2 - 3x_3 &= -2x_3 +\end{matrix} &\rightarrow x_1 + x_2 =0 +\end{align*} +On choisit donc $x_3$ quelconque et $x_1=-x_2$ ce qui correspond à un sous espace propre de dimension 2 et : +\begin{align*} +Ker\{ \lambda_0 \mathbb{1}_3 - A\} &= Vect \left\{ \begin{pmatrix} +1\\-1\\0 +\end{pmatrix} , \begin{pmatrix} +0\\0\\1 +\end{pmatrix} \right \} +\end{align*} +La dimension de ce sous espace propre est $2 \leq 3$ donc A est non diagonalisable.\\ +On a donc deux blocs de Jordan car la multiplicité des valeurs propres de 2 : +\[J=\begin{pmatrix} +\lambda_0 & 1 & 0\\ +0 & \lambda_0 & 0\\ +0 & 0 & \lambda_0 +\end{pmatrix} \textbf{ ou alors } \begin{pmatrix} +\lambda_0 & 0 & 0\\ +0 & \lambda_0 & 1\\ +0 & 0 &\lambda_0 +\end{pmatrix} = J\] + +Le but maintenant est de trouver un 3ème vecteur pour compléter la bases de vecteurs propres et avoir $V = \begin{pmatrix}v_1 & v_2 & v_3 \end{pmatrix}$ tel que $V^{-1}AV = J$, avec J l'une des deux matrices contenant un bloc de Jordan.\\ +En posant AV = VJ on a : +\begin{align*} +\begin{pmatrix} +Av_1 & Av_2 & Av_3 +\end{pmatrix}&= \begin{pmatrix} +\lambda_0v_1 & \lambda_0 v_2 & v_2 + \lambda_0 v_3 +\end{pmatrix} +\intertext{et on obtient le système: } +\begin{matrix} +Av_1\\ +Av_2\\ +Av_3 +\end{matrix} &= \begin{matrix} +\lambda_0v_1\\ +\lambda_0 v_2\\ +v_2 + \lambda_0 v_3 +\end{matrix} +\intertext{on obtient donc en particulier pour le vecteur $v_3$ :} +\begin{pmatrix} +1 & 1& 0\\ +-1& -1 & 0\\ +1& 1& 0 +\end{pmatrix} v_3 = \begin{pmatrix} +0\\0\\1 +\end{pmatrix} +\end{align*} +On constate qu'il n'est pas possible de déterminer $v_3$ de cette façon, on cherche donc a prendreà la place de $v_2$, $\tilde{v_2} = \alpha v_1 + \beta v_2$ de sorte à avoir $A\tilde{v_2} = \lambda_0 \tilde{v_2}$ (ce qui reste vrai car on prend une combinaison liéaire de 2 vecteur propre du ker) et surtout $(A-\lambda_0 \mathbb{1}_3)v_3 = \tilde{v_2}$. On a donc : +\begin{align*} +\begin{pmatrix} +1 & 1& 0\\ +-1& -1 & 0\\ +1& 1& 0 +\end{pmatrix} v_3 &= \alpha \begin{pmatrix} +1\\-1\\0 +\end{pmatrix} + \beta \begin{pmatrix} +0\\0\\1 +\end{pmatrix} +\intertext{avec , $\beta = \alpha = 1$} +\tilde{v_2} &= \begin{pmatrix}1\\-1\\1\end{pmatrix} +\intertext{on résoud alors pour trouver $v_3$ :} +v_3 = \begin{pmatrix}0\\1\\1\end{pmatrix} +\intertext{on a donc finalement trouvé V :} +V &= \begin{pmatrix} +1&1&0\\-1&-1&1\\0&1&1 +\end{pmatrix}\\ +V^{-1} &= \begin{pmatrix} +2&1&-1\\-1&-1&1\\1&1&0 +\end{pmatrix}\\ +J &= V^{-1}AV = \begin{pmatrix} +-2&0&0\\0&-2&1\\0&0&-2 +\end{pmatrix} +\end{align*} +On a bien une matrice diagonale par bloc avec un bloc de Jordan. +On pose $\xi \in \mathbb{R}^3$, $x = V\xi$, et on a le système : +\begin{align*} +\left \{ \begin{matrix} +\dot{x} = Ax + Bu & x_0 \in \mathbb{R}^n\\ +y = Cx + Du & +\end{matrix} \right. &\rightarrow \left \{ \begin{matrix} +\dot{\xi} = J\xi + V^{-1}B u &= J\xi + \tilde{B}u\\ +y = CV \xi +D u &= \tilde{C}\xi + \tilde{D} u +\end{matrix} \right. +\intertext{avec, } +\tilde{B} &= \begin{pmatrix}-1\\1\\1\end{pmatrix}\\ +\tilde{C} &= \begin{pmatrix}0 &1&0\end{pmatrix}\\ +\tilde{D} &= 0 +\end{align*} +\bigbreak +\noindent 2- \begin{align*} +\dot{\xi} &= \begin{pmatrix}-2&0&0\\0&-2&1\\0&0&-2\end{pmatrix}\xi + \begin{pmatrix}-1&1&1\end{pmatrix}\\ +y &= \begin{pmatrix}0&1&0\end{pmatrix} \xi\\ +C(A,B) &= \begin{pmatrix}0&1&-4\\1&-3&8\\2&-3&4\end{pmatrix}\\ +det(C(A,B)) &= 0 +\end{align*} +Non commandable. +\end{document} diff --git a/421-Controle_processus/TD7/TD7.pdf b/421-Controle_processus/TD7/TD7.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a4e223c383451193c848c6dfb9b3ce64668bceb1 GIT binary patch literal 183191 zcma&NV{l}D7_A%IPA1mGwrz7_Ta!#|+qP|IqKR$W$xLkX?*F+wbRo&IKE4_Q~ zAKvw>^>!$f#3bmM8Q9?{W*3K6;kbyIh#icriTU{880F0DEu1ZhS(usFiT}p~j#0wO z*4fO7m{G#k$k|NH%*4Ue3{F4*?w7NZnUNiw`}&2pPTU3umhV%YQl5qcO=_4M)P-o$ 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+\begin{document} + +\titre{\nom} + + +\subsection*{Exercice 1 :} +On considère le système suivant : +\[(S)= \left \{ + \begin{matrix} + \dot{x} = \begin{pmatrix}-10 & 12\\-4 & 7\end{pmatrix} x_1 + \begin{pmatrix} -1\\-1\end{pmatrix} u\\ + y_1 =\begin{pmatrix}4 & 5\end{pmatrix}x_1 + \end{matrix} + \right. +\] +\begin{enumerate} +\item +\begin{enumerate} +\item Base modale \\ +Déterminons les valeurs propres de la matrice d'évolution :\\ +\begin{align*} +P(\lambda) &= det(A_1 - \lambda \mathbf{1}_2)\\ +&= det \begin{pmatrix}-10-\lambda & 12 \\ -6 & 7 - \lambda \end{pmatrix}\\ +&= (-10-\lambda)(7-\lambda)+72\\ +&= -70 + 3\lambda + \lambda^2 +72\\ +&= \lambda^2 + 3 \lambda +2\\ +&= (\lambda + 1 )(\lambda + 2) +\end{align*} + + +Cherchons les vecteurs propres vérifiant $A_1X = \lambda X$:\\ +Pour $\lambda = -1$ : +\begin{align*} +A_1X = \lambda X &\rightarrow -6x_1 + 8x_2 =0 +\end{align*} +On a donc : +\begin{align*} +E_{-1} &= Ker\{ -1. \mathbf{1}_3 - A\} \\ +&= Vect\left \{ \begin{pmatrix} +4\\3 +\end{pmatrix}\right \} +\end{align*} +\bigbreak +Pour $\lambda = -2$ : +\begin{align*} +A_1X = \lambda X &\rightarrow -6x_1 + 9x_2 =0 +\end{align*} +On a donc : +\begin{align*} +E_{-2} &= Ker\{ -2. \mathbf{1}_3 - A\} \\ +&= Vect\left \{ \begin{pmatrix} +3\\2 +\end{pmatrix}\right \} +\end{align*} +\bigbreak +La matrice de changement de base est donc : $P = \begin{pmatrix}4&3\\3&2\end{pmatrix}$\\ +\item Le système est globalement asymptotiquement stable car les valeurs propres sont à $Re() < 0$.\\ +On effectue alors le changement de base $x_1 = P \xi_1$ + +\begin{align*} +\left \{ \begin{matrix} +\dot{x}_1 = A_1+x_1 +B_1 u_1\\ +y_1 = C_1 x_1 +\end{matrix} \right. \rightarrow \left \{ \begin{matrix} +\dot{\xi}_1 = \Lambda\xi_1 + V^{-1}B_1u_1\\ +y_1 = C_1V\xi_1 +\end{matrix} \right.\\ +\intertext{avec,} \Lambda = V^{-1}A_1V = \begin{pmatrix} +-1&0\\0&-2 +\end{pmatrix} +\intertext{On a alors :} +\left \{\begin{matrix} +V^{-1}B_1 = \begin{pmatrix}-1\\1 \end{pmatrix} = B_m\\ +C_1V = \begin{matrix}1&2\end{matrix} = C_m +\end{matrix} \right. +\end{align*} +\bigbreak + +\item Rappel : pour $\dot{x} = Ax +Bu$ avec $x(0)=x_0 \in \mathbb{R}^n$ la solution est de la forme CI + régime forcé : +\[ x(t) = e^{At}x_0 + \int_0^te^{A(t-\tau)}Bu(\tau)d\tau\] +\bigbreak +Application : +\begin{align*} +\xi_1(t) &= e^{\Lambda t}\xi_0 + \int_0^te^{\Lambda (t-\tau)}B_mu_1(\tau)d\tau\\ +e^{\Lambda t} &= e^{\begin{pmatrix}-t&0\\0&-2t\end{pmatrix}} = \begin{pmatrix} +e^{-t}&0\\0&e^{-2t}\end{pmatrix}\\ +\xi_1(t) &= \begin{pmatrix} +\int_0^t -e^{-(t-\tau)}Bu(\tau)d\tau\\ +\int_0^t e^{-2(t-\tau)}Bu(\tau)d\tau +\end{pmatrix}\\ +y_1(t) &= C_m\xi_1(t)\\ +&= \int_0^t -e^{-(t-\tau)}Bu(\tau)d\tau + \int_0^t 2e^{-2(t-\tau)}Bu(\tau)d\tau +&= \int_0^t (2e^{-2(t-\tau)}-e^{-(t-\tau)})Bu_1(\tau)d\tau +\end{align*} +Pour une réponse indicielle $u_1(t) = 1$ $\forall t \geq 0$\\ + +\item Commandabilité : +C(A,B) = $\begin{pmatrix}B & AB \end{pmatrix}$ +\begin{align*} +A &= V\Lambda V^{-1}\\ +B &= VB_m +\intertext{donc :} +C(A,B) &= \begin{pmatrix}VB_m & V\Lambda V^{-1}VB_m \end{pmatrix}\\ +&= \begin{pmatrix} VB_m & V\Lambda B_m \end{pmatrix}\\ +&= V \begin{pmatrix}B_m & \Lambda B_m \end{pmatrix}\\ +&= V C(\Lambda,B_m) +\intertext{or,} +C(\Lambda,B_m) &= \begin{pmatrix}-1&1\\1&-2\end{pmatrix}\\ +\text{donc, } det (C(\Lambda,B_m)) &= 1 \neq 0 +\end{align*} +Ainsi, le système est commandable.\\ + +Remarque : Soit $x_1 \in \mathbb{R^2}$: +\begin{align*} +x_1(t) &= e^{A_1t}x_0 + \int_0^t e^{A_1(t-\tau)}B_1u_1(\tau)d\tau \in \mathbb{R^2}\\ +W_c(0,t) &= \int_0^t e^{A_1(t-\tau)}B_1B_1^Te^{A_1^T(t-\tau)}d\tau \in \mathbb{R^{2*2}}\\ +\intertext{$W_c(0,t_1)$ est inversible car commandable} +x_1(t_1) &= e^{A_1t_1}x_0 + \int_0^{t_1} e^{A_1(t_1-\tau)}B_1B_1^Te^{A_1^T(t_1-\tau)}W_c(0,t_1)^{-1}d\tau (x_1 - e^{A_1t_1}x_0) =x_1 \forall t_1 \geq 0\\ +&= B_1e^{A(t_1-t)}W_c(0,t_1)^{-1}(x_1-e^{A_1t_1}x_0) +\end{align*} +\bigbreak +\end{enumerate} + + +\item On considère le système (S2) suivant : +\[(S2)= \left \{ + \begin{matrix} + \dot{x}_2 = -10x_2 + 4u_2 & x_2(0) = x_0 \in \mathbb{R}^n\\ + y = -2x + u_2 & + \end{matrix} + \right. +\] + +\begin{center} +\includegraphics[scale=0.7]{TD7.png} +\end{center} +\bigbreak +\begin{enumerate} +\item La relation de connection est $u_1 = y_2$ et donne +\[\dot{\xi}_1 = \Lambda \xi_1 + B_m u_1 \text{ avec, } u_1 = y_2 = C_2x_2+ D_2u_2\] +D'où le système suivant : +\begin{align*} +\left \{ \begin{matrix} +\dot{\xi}_1 = \Lambda \xi_1 + B_mC_2x_2 + B_mD_2u_2\\ +\dot{x}_2 = A_2x_2 + B_2u_2 +\end{matrix} \right. +\end{align*} +Posons x(t) = \begin{pmatrix}\xi_1\\x_2\end{pmatrix}, on a alors : + +\begin{align*} +\dot{x} &= \begin{pmatrix} +\Lambda & B_mC_2\\ +0 & A_2 +\end{pmatrix} x + \begin{pmatrix} +B_mD_2\\B_2 +\end{pmatrix} u\\ +\dot{x} &= \begin{pmatrix} +-1 & 0 & 2\\ +0 & -2 & -2\\ +0 & 0 &-10 +\end{pmatrix} x + \begin{pmatrix} +-1 \\ 1 \\4 \end{pmatrix} u = Ax + Bu +\intertext{On a donc pour l'équation d'observation :} +y &= y_1 = C_m\xi_1 + D_1u_1\\ +&= C_m \xi_1 + D_1y_2 \text{ où }D_1 = 0\\ +&= C_m\xi_1\\ +&= \begin{pmatrix} +1 & 2 & 0 \end{pmatrix}.\begin{pmatrix} +\xi_1 \\ x_2 +\end{pmatrix} +\end{align*} + +\item On regarde alors la commandabilité : +\begin{align*} +C(A,B) &= \begin{pmatrix} B & AB & A^2B \end{pmatrix}\\ +&= \begin{pmatrix} +-1 & 9 & -89 \\ +1 & -10 & 100\\ +4 & -40 & 400 +\end{pmatrix}\\ +rang(C(A,B)) &= 2 \text{ car } L_3 = 4L_2 +\end{align*} + +(S1) est stable, (S2) est stable mais l'association des deux ne l'est pas !\\ + +\item On considère le système (S) suivant : +\[(S)= \left \{ + \begin{matrix} + \dot{x} = Ax + Bu & x(0) = x_0 \in \mathbb{R}^n\\ + y = Cx + Du & + \end{matrix} + \right. +\] +\noindent On introduit le vecteur : +\begin{align*} +\dot{x} &= \begin{pmatrix} +\dot{x}_1\\ +\dot{x}_2\\ +\vdots +\dot{x}_n\\ +\end{pmatrix} +\intertext{et, sa transformée de Laplace} +L\{x\} &= \begin{pmatrix} +L\{x_1\}\\ +L\{x_2\}\\ +\vdots\\ +L\{x_n\}\\ +\end{pmatrix}\\ +&= \begin{pmatrix} +X_1(p)\\ +X_2(p)\\ +\vdots\\ +X_n(p)\\ +\end{pmatrix} = X(p) +\intertext{on a alors avec le système (S)} +pX(p) -x_0 &= AX(p)+ BU(p)\\ +(p\mathbf{1_n}-A)X(p) &= x_0 +BU(p) +\intertext{d'où la relation :} +X(p) &= (p\mathbf{1_n}-A)^{-1}BU(p) + (p\mathbf{1_n}-A)^{-1}x_0 +\intertext{De même, on pose le vecteur :} +Y(p) &= L\{y(t)\}\\ +&= CX(p) + DU(p)\\ +&= [C(p\mathbf{1_n}-A)^{-1}B+D]u(p) + C(p\mathbf{1_n}-A)^{-1} x_0\\ +&= G(p) U(p) + C(p\mathbf{1_n}-A)^{-1} x_0\\ +\intertext{et on a :} +(p\mathbf{1_n}-A)^{-1} &= \frac{1}{det(p\mathbf{1_n}-A)} Adj(p\mathbf{1_n}-A)\\ +G(p) &= \frac{C.Adj(p\mathbf{1_n}-A)B + DP_A(p)}{P_A(p)} +\end{align*} +Les pôles sont les modes. +On calcul donc : +\begin{align*} +p\mathbf{1_n}-A &= \begin{pmatrix} +p+1 & 0 & -2\\ +0 & p+2 & 2\\ +0 & 0 & p+10 +\end{pmatrix} +\intertext{puis,} +(p\mathbf{1_n}-A)^{-1} &= \frac{1}{(p+1)(p+2)(p+10)}\begin{pmatrix} +(p+2)(p+10) & 0 & 0\\ +0 & (p+1)(p+10) & 0\\ +2(p+2) & -2(p+1) & (p+1)(p+2) +\end{pmatrix}^T +\intertext{ensuite,} +C(p\mathbf{1_n}-A)^{-1} &= \frac{1}{(p+1)(p+2)(p+10)} \begin{pmatrix} +(p+2)(p+10) & 2(p+1)(p+10) & 2(p+2)-4(p+1) +\end{pmatrix} +\intertext{enfin on obtient l'ordre 2 :} +C(p\mathbf{1_n}-A)^{-1}B &= G(p) = \frac{p}{(p+1)(p+10)} +\end{align*} +Remarque : +\begin{align*} +G(p) &= G_1(p) G_2(p) +\intertext{avec} +G_1(p) &= C_1(p\mathbf{1_2}-A_1)B_1 = \frac{p}{(p+1)(p+2)}\\ +G_2(p) &= C_2(p\mathbf{1_1}-A_2)B_2 + D_2 = \frac{p+2}{p+10} +\end{align*} +Un pôle de $G_1$ a été neutralisé par un zéro de $G_2$, on a donc une perte de commandabilité.\\ +Réalisation minimale : un vecteur d'état de taille la plus petite. Ici, ordre 2 (on avait un ordre 3 qui n'était pas minimal). +\end{enumerate} +\item On considère le système suivant :\\ +\[\left \{ \begin{matrix} +\dot{x_1} &= &\begin{pmatrix}-10&12\\-6&7\end{pmatrix} x_1 &+ \begin{pmatrix}-1\\-1\end{pmatrix}\\ +y_1 &= &\begin{pmatrix}4&-5\end{pmatrix}x_1 +\end{matrix} \right.\] +\begin{enumerate} + +\item On impose une trajectoire polynomiale : $y_d(t) = \sum_{k=0}^n \alpha_k \left(\frac{t}{T}\right)^k$\\ +Avec les conditions initiales suivantes :\\ \smallbreak +\begin{center} +\begin{tabular}{|c|c|} +\hline +t=0 & t=T \\ +\hline +\hline +$y_1(0)=0$ & $y_1(T)=1$ \\ +\hline +$\dot{y_1}(0)=0$ & $\dot{y_1}(T)=0$ \\ +\hline +$\ddot{y_1}(0)=0$ & $\ddot{y_1}(T)=0$ \\ +\hline +\end{tabular} +\end{center} + +On a aussi : +\begin{align*} +\dot{y_1}(t) &= \sum_{k=0}^n \frac{k}{T}\alpha_k \left(\frac{t}{T}\right)^{k-1}\\ +\ddot{y_1}(t) &= \sum_{k=0}^n \frac{k(k-1)}{T^2}\alpha_k \left(\frac{t}{T}\right)^{k-2}\\ +\end{align*} +\bigbreak + +On a 6 contraintes donc 6 inconnues donc au maximum, $n=5$ + +\paragraph{à $t = 0$} :\\ +\begin{align*} +y_1(0) &= \sum_{k=0}^5 \alpha_k \left(\frac{0}{T}\right)^k \Leftrightarrow \alpha_0 = 0\\ +\dot{y_1}(0) &= \sum_{k=0}^5 \frac{k}{T}\alpha_k \left(\frac{0}{T}\right)^{k-1} \Leftrightarrow \alpha_1 = 0\\ +\ddot{y_1}(t) &= \sum_{k=0}^5 \frac{k(k-1)}{T^2}\alpha_k \left(\frac{0}{T}\right)^{k-2} \Leftrightarrow \alpha_2=0\\ +\end{align*} + +\paragraph{à $t = T$} : +\begin{align*} +y_1(T) = 1 &\Leftrightarrow \sum_{k=0}^5 \alpha_k = 1 \\ +&\Leftrightarrow \alpha_3+\alpha_4 + \alpha_5 = 1\\ +\dot{y_1}(T)=0 &\Leftrightarrow \sum_{k=3}^5 \frac{k}{T}\alpha_k = 1 \\ +&\Leftrightarrow \frac{1}{T}(3\alpha_3 + 4\alpha_4 + 5 \alpha_5) = 0\\ +\ddot{y_1}(T) = 0 &\Leftrightarrow \sum_{k=3}^5 \frac{k(k-1)}{T^2}\alpha_k = 0 \\ +&\Leftrightarrow \frac{1}{T^2}(6\alpha_3 + 12\alpha_4 + 20\alpha_5) = 0\\ +\intertext{ce système est alors équivalent au calcul matriciel suivant :} +\begin{pmatrix}1&1&1\\3&4&5\\6&12&20\end{pmatrix}.\begin{pmatrix}\alpha_3\\ \alpha_4 \\ \alpha_5\end{pmatrix} &= \begin{pmatrix}1\\0\\0 \end{pmatrix}\\ +\begin{pmatrix}\alpha_3\\ \alpha_4 \\ \alpha_5\end{pmatrix} &= \begin{pmatrix}1&1&1\\3&4&5\\6&12&20\end{pmatrix}^{-1}.\begin{pmatrix}1\\0\\0 \end{pmatrix} +\end{align*} + +Le calcul abouti à : +\[\left \{\begin{matrix} +\alpha_3 = 10\\ +\alpha_4 = -15\\ +\alpha_5 = 6 +\end{matrix} \right.\] + +Remarque : On a une matrice 3x3, on peut donc se permettre de calculer l'inverse à partir des cofacteurs $\Delta_{ij} = (-1)^{i+j}|M_{ij}|$ avec $M_{ij}$ la matrice obtenu en supprimant la ligne i et la colonne j.\\ + +Remarque : $\ddot{y_1}(t)$ représente la secousse, aussi appelé Jerk. +Le quintique est la trajectoire à Jerk minimal. +\bigbreak +\bigbreak +\item Passage à la forme canonique de commandabilité.\\ +Il s'agit de trouver la matrice de passage M du système d'état vers celui correspondant a $A_c$ une matrice compagnon horizontale de type 1 et $B_c$ un vecteur de 0 avec 1 sur la dernière composante. Puis, une fois que l'on à M, on calcule $C_c = M.C$\\ + +On a le polynôme caractéristique $P_{A_1}(\lambda) = \lambda^2 + 3\lambda + 2 = \lambda^2 + a_1\lambda + a_0$, dont on en déduit la matrice compagnon horizontale : +\[ A_c = \begin{pmatrix} +0&1\\-a_0&-a_1 +\end{pmatrix} = \begin{pmatrix} +0&1\\-2&-3 +\end{pmatrix}\] +Et on a aussi : +\[B_c = \begin{pmatrix} +0\\1\end{pmatrix}\] + +Changement de coordonnées $M \in \mathbb{R}^{2x2}$ avec $M = \begin{pmatrix}m_1&m_2\end{pmatrix}$ +\begin{align*} +M^{-1}AM = A_c &\Leftrightarrow AM = MA_c\\ +&\Leftrightarrow\begin{pmatrix} +Am_1& Am_2 +\end{pmatrix} = \begin{pmatrix} +m_1&m_2 +\end{pmatrix}.\begin{pmatrix} +0&1\\-2&-3 +\end{pmatrix}= \begin{pmatrix} +-2m_2& m_1 - 3m_2 +\end{pmatrix}\\ +&\Leftrightarrow \left\{ \begin{matrix} +Am_1 = -2m_2\\ +Am_2 = m_1 - 3m_2 +\end{matrix}\right.\\ +&\Leftrightarrow \left \{ \begin{matrix} + Am_1 = -2m_2\\ + (A+3\mathbf{1}_2)m_2 = m_1 +\end{matrix}\right. +\intertext{or, $M^{-1}B = B_c\Leftrightarrow B=MB_c=m_2$, donc :} +M &= \begin{pmatrix} +B & (A+3\mathbf{1}_2)B +\end{pmatrix} +\intertext{d'où : } +M &= \begin{pmatrix}-5&-1\\-4&-1\end{pmatrix} +\end{align*} + +Il est inutile de calculer $M^{-1}$ pour le calcul de la forme canonique. Car $C_c = C_1M = \begin{pmatrix}0&1\end{pmatrix}$ et, $D_c = D_1 = 0$ +\item On impose $y_1(t) = 10\left(\frac{t}{T}\right)^3 - 15\left(\frac{t}{T}\right)^4 + 6\left(\frac{t}{T}\right)^5$.\\ +On cherche une commande du type : $u_1(t) = \sum_{k=0}^m \beta_k \left(\frac{t}{T}\right)^k$.\\ + +\end{enumerate} +\paragraph{Forme de Browmovski }: + +\begin{align*} +\begin{pmatrix} +\dot{z_1}\\\vdots\\\vdots\\\dot{z_n}\end{pmatrix} = \begin{pmatrix} +0&1&0&...&0\\ +0&0&\ddots&&\\ +0&...&...&1\\ +-a_0&...&...&-a{n-1} +\end{pmatrix}.\begin{pmatrix} +z_1\\\vdots\\\vdots\\z_n +\end{pmatrix} + \begin{pmatrix} +0\\\vdots\\0\\1 +\end{pmatrix}u +\end{align*} + +Ayant la forme canonique de commandabilité, la forme de Browmovski conciste à poser :\\\[\boxed{v(t) = -a^Tz + u}\] +où $a = \begin{pmatrix} +a_0 &...&a_{n-1} +\end{pmatrix}^T$ +d'où +\begin{align*} +\begin{pmatrix} +\dot{z_1}\\\vdots\\\vdots\\\dot{z_n}\end{pmatrix} = \begin{pmatrix} +0&1&0&...&0\\ +0&0&\ddots&&\\ +0&...&...&1\\ +0&...&...&0 +\end{pmatrix}.\begin{pmatrix} +z_1\\\vdots\\\vdots\\z_n +\end{pmatrix} + \begin{pmatrix} +0\\\vdots\\0\\1 +\end{pmatrix}v +\end{align*} +Avec l'équation d'observation du système d'état $y = \begin{pmatrix} +c_0&...&c_{n-1} +\end{pmatrix}.\begin{pmatrix} +z_1\\\vdots\\z_{n-1} +\end{pmatrix}$ + +\paragraph{Application }: +Avec l'équation d'observation : +\begin{align*} +y_1 &= \begin{pmatrix}0&1\end{pmatrix} . \begin{pmatrix}z_1\\z_2\end{pmatrix} = z_2 +\intertext{Puis, avec la forme de Brow***:} +&\left \{\begin{pmatrix} +\dot{z_1} = z_2\\ +\dot{z_2} = v +\end{pmatrix}\right. \text{où $v= -\begin{pmatrix} +2&3 +\end{pmatrix}\begin{pmatrix} +z_1\\z_2 +\end{pmatrix} + u$} +\intertext{On a donc comme commande, en remplaçant avec les expressions provenant de l'équation d'obersation :} +u&= v + \begin{pmatrix} +2&3 +\end{pmatrix}\begin{pmatrix} +z_1\\z_2 +\end{pmatrix} +&=2z_1 + 3y_1 + \dot{y_1}\\ +\intertext{On calcul donc chaque terme :} +z_1 &= \int_0^tz_2(\tau) d\tau\\ +&= \int_0^ty_1(\tau) d\tau\\ +&= \frac{10T}{4}\left(\frac{t}{T}\right)^4 -\frac{15T}{5}\left(\frac{t}{T}\right)^5 + \frac{6T}{6}\left(\frac{t}{T}\right)^6 + cst(=0) +\intertext{donc :} +u(t) &=2z_1 + 3y_1 + \dot{y_1}\\ +&= 2T\left(\frac{t}{T}\right)^6 + (18-6T)\left(\frac{t}{T}\right)^5 + (\frac{30}{T}-45 + 5T) \left(\frac{t}{T}\right)^4 + (30-\frac{60}{T})\left(\frac{t}{T}\right)^3 + \frac{30}{T}\left(\frac{t}{T}\right)^2\\ +&= \beta_6\left(\frac{t}{T}\right)^6 + \beta_5 \left(\frac{t}{T}\right)^5 + ...+\beta_2 \left(\frac{t}{T}\right)^2 +\end{align*} + +Donc m=6 et $\beta_1 = \beta_2 = 0$. Commande en boucle ouverte non robuste au conditions initiales. 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\dot{\epsilon_d}(0^+) + h_1(p\epsilon_d - \epsilon_d(o^+))+ h_0\epsilon_d(p) = 0\\ +\epsilon_d &= \frac{\dot{\epsilon_d}(0^+)+(h_1+p)\epsilon_d(0^+)}{p^2+h_1p + h_0}\\ +&=\frac{\gamma_1p+\gamma_0}{(p-p_1)(p-p_2)} +\end{align*} +Exemple : $p^2+h_1p + h_0 = (p+\frac{10}{T})^2 = p^2 + \frac{20p}{T} + \frac{100}{T^2}$\\ + +$v$ doit vérifier d'après l'expression de $\ddot{\epsilon_d}$ : +\begin{align*} +v(t) &= \ddot{\epsilon_d}(t) + \ddot{z_1^d}(t)\\ +&= \ddot{z_1^d}(t) - h_1(\dot{z_1}(t)-\dot{z_1^d}(t))- h_0(z_1(t) - z_1^d(t))\\ +&= \begin{pmatrix} +h_0&h_1\end{pmatrix}.\begin{pmatrix} +z_1\\z_2\end{pmatrix} + (\ddot{z_1^d}(t) + h_1\dot{z_1^d}(t)+ h_0z_1^d(t))\\ +&= -Kx_c + \tilde{z_d} +\intertext{or, $v=-a^Tx_c + u$, donc on obtient la loi de commande :} +u &= -Kx_c + \tilde{z_d} + a^Tx_c\\ +&= (a^T-K)x_c + \tilde{z_d} +\end{align*} + +Loi de commande en BF par retour d'état : +\begin{center} +\includegraphics[scale=0.5]{TD8.png} +\end{center} + + + +\end{enumerate} +\end{enumerate} + +\section{Exercice 2 : Synthèse d'une loi de commande par retour d'état} +\begin{enumerate} +\item On a ici, n=3. Déterminons la représentation d'état.\\ +On a avec la fonction de transfert :\\ +\begin{align*} +A(p)Y(p) &= B(p)U(p)\\ +\intertext{d'où l'équation dans le domaine temporelle :} +y^{(3)} + 8 y^{(2)} + 17 y^{(1)} + 10y &= u^{(1)} +2u +\intertext{On a donc la forme canonique de commandabilité :} +\dot{x_c} &= \begin{pmatrix}0&1&0\\0&0&1\\-10&-17&-8\end{pmatrix} x_c + \begin{pmatrix}0\\0\\1\end{pmatrix}u\\ +&= A_c x_c +B_c u +\intertext{et on a l'equation d'observation :} +y &= \begin{pmatrix}2\\1\\0\end{pmatrix} x_c + 0.u +\end{align*} +Attention : Le u de l'équation d'état et de la fonction de transfert ne sont PAS les mêmes, l'un est un scalaire l'autre un vecteur.\\ + +On a une forme canonique de commandabilité, donc le système est effectivement commandable. Cependant, l'observabilité n'est pas acquise.\\ +La réalisation est observable ssi il n'y a pas de simplification d'un zéro par un pôle. Il suffit donc de vérifier que -2 n'est pas un zéro du numérateur.\\ +Une représentation minimal est appelée réalisation d'état.\\ + +\item On impose pour la boucle fermée : +\[ \left \{ \begin{matrix} +u = -Kx_c + \eta e\\ +K = \begin{pmatrix}k_0 & k_1 & k_2\end{pmatrix} \in \mathbb{R^{1x3}}\\ +\end{matrix} \right. \] + +On a donc pour l'équation d'état : +\begin{align*} +\dot{x_c} &= A_c x_c + B_c u\\ +&= (A_c - B_cK)x_c + \eta B_c e\\ +&= A_{bf}x_c + \eta B_c e +\intertext{et pour l'éuation d'obersation : } +y = C_c x_c +\end{align*} + +\item On impose les racines $P_1/P_2 = -m\omega_0 \pm j \omega_0\sqrt{1-m^2}$, donc on a le polynôme caractéristique : $\frac{p^2}{\omega_0^2} + \frac{2m}{\omega_0}p + 1$\\ +Il est nécessaire de spécifier un troisième pôle $p_3 = -\lambda m \omega_0$, avec $\lambda >> 1$, car la système est d'ordre 3. Comment le choisir? Stable, plus rapide que les autres pôles que l'on impose.\\ + +On a alors le polynôme à imposer : +\begin{align*} +\Pi_d(p) &= (p + \lambda m \omega_0)(p^2 + 2mp + \omega^2)\\ +&= p^3 + (2+\lambda)m\omega_0 p^2 + (2\lambda m^2 + 1) \omega^2 p + \lambda m \omega_0^3 +\end{align*} + +Pour la matrice $A_{bf}$, on a le polynôme caractéristique : +\begin{align*} +P_{A_{bf}} &= det(p\mathbf{1_3} - A_{bf})\\ +&=p^3 + (8+k_2)p^2 + (17+k_1)p + (10+k_0) +\end{align*} +Sachant qu'il y a une lien directe entre les coefficients de la matrice compagnon et son polynôme caractéristique.\\ +On identifie donc les coefficients des deux polynômes : +\[\left \{ \begin{matrix} +(8+k_2) = (2+\lambda)m\omega_0\\ +(17+k_1) = (2\lambda m^2 + 1) \omega^2\\ +(10+k_0) = \lambda m \omega_0^3 +\end{matrix} \right. \] + +\item Erreur statique nulle ssi le gain en p=0 vaut 1 (gain statique unitaire)\\ +Or, \[G_{bf}(p) = C_c(p \mathbf{1_3}-A_{bf})^{-1}B_c \eta \] +Donc , \[ \eta = \frac{1}{-C_c(A_c-B-cK)^{-1}B_c}\] +\end{enumerate} + +\end{document} diff --git a/421-Controle_processus/TD9/TD9.pdf b/421-Controle_processus/TD9/TD9.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5fdb1d7bce289796f38f52b18581302dba23c80f GIT binary patch literal 122446 zcma(2W2`Vt&}fNn+qP}nwr$(CZQItfZQHhO+n6`^&NrFloMcY_s{U8$PP$h2swPzs z5u;_KV}&A}UmjV9;virkus5 zJ&>$SJ#CGdaXiVm+`2hKc5Nc<>u9sl@nk!ci7GDFOO%af5B>Q=d~|*tz3-2^R)=BB z+UTi0XyM3}P6jOmZJk{d`zimPq#Rl*LJ`dmbWOeA8PR;vk54P@J%2q4RXX>VU%#&R zo5%8K^>p-nIt;WzQ10m}97Y-RwKjBn|Kbr?zpu)IVNTuea!gNc37C`3!=FwM=Z7g! zSZrJ`TNr~XgCvN`4)#E4$sDE71E{@;)hM?)9d_?ivMzI(lA}l}Gm^qp)YPQ)J_~cN zm(t!oJ>9j8v5KS?`nj!icKx<4Q1uh;$qyk&<<&OB?eNh3leO%x6ThC%=DMvDqcLOM z@ENu3zD~EwHTQqPcj@^2bX0azUOly2b*ocsXjd~;KZj)B_rkvZZ5wrK)vs@70}N?2 zBM~i7zt(D=;-K)w7KM?_gP)3YyF;m+-}c-rdT5MIavsj5%89ubRaq9)Q5Mt3MfmdRLO2-tT99%GWO^50 zI?o3nAm&`}W@7i|BYF9wdJNJbi1LWu1DHiz^zT-3*B{^OQzS&+hz)1R4d}|$m_|C? z^Hcrc`LBVHc5LqMu4b+?PSnBa6zLEGpdtWfrRN_H*ToOWfcB<+k12hAg|DBwa$Ew{cPj!^;5~IwiHT|X)mw# 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z_;)|F?I2ZM<2T6)!?bDM?LY^l5O?I$63*siuB^ zpPy9+JR8Y6j>6D2hdcMC2g4XTLS$epXZGp`n;BaCL|ZPNh6jzpz&CvkO%p#7KZ0Z$ f{`)KI=w#&J=latex,->] (-6,0) -- (6,0) node[right]{$f$} ; +\draw [>=latex,->] (0,-2) -- (0,2) node[left]{$|X(f)|$}; +\draw [red] (1,0)node[below]{$f_0$} -- (1,1); +\draw [red] (-1,0)node[above]{$-f_0$} -- (-1,-1); +\draw (5,0)node[below]{$f_0+F_e$} -- (5,1); +\draw (3,0)node[above]{$-f_0+F_e$} -- (3,-1); +\draw (-3,0)node[below]{$f_0-F_e$} -- (-3,1); +\draw (-5,0)node[above]{$-f_0-F_e$} -- (-5,-1); +\draw [dotted] (-6,1) node[left]{$1/T_e$} -- (6,1); + +\end{tikzpicture} +\caption{Représentation du spectre $X^*(f)$ dans le cas où $F_e > 2f_0$} +\end{figure} + +\item Pour la reconstruction du signal, on utilise un filtre passe-bas idéal : +\[ H(p=j2\pi f) = \left\{ +\begin{array}{ll} +1 & \si |f| \leq F_e/2 \\ +0 & \sinon +\end{array} +\right. +\] + +\begin{itemize} +\item Si $f_0 < F_e / 2$, alors on va couper toutes les composantes sauf celles en $f_0$ et en $-f_0$. Le signal reconstitué sera donc le signal sinusoïdal initial. +\item Si $f_0 = 1,2F_e$, alors le spectre sera de la forme donnée en figure 2. +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\draw [>=latex,->] (-4,0) -- (5,0) node[right]{$f$} ; +\draw [>=latex,->] (0,-2) -- (0,2) node[left]{$|X(f)|$}; +\draw [red] (1,0)node[below]{$f_0$} -- (1,1); +\draw [red] (-1,0)node[above]{$-f_0$} -- (-1,-1); +\draw (4.2,0)node[below]{$f_0+F_e$} -- (4.2,1); +\draw (2.2,0)node[above]{$-f_0+F_e$} -- (2.2,-1); +\draw (-0.2,0)node[below]{$f_0-F_e$} -- (-0.2,1); +\draw (-2.2,0)node[above]{$-f_0-F_e$} -- (-2.2,-1); +\draw [dotted] (-4,1) node[left]{$1/T_e$} -- (5,1); +\end{tikzpicture} +\caption{Représentation du spectre $X^*(f)$ dans le cas où $F_e = 1.2f_0$} +\end{figure} + +Ainsi, en utilisant le filtre passe-bas, on ne récupérera que la composante correspondant à la fréquence $f_0-F_e$, et la reconstitution sera donc incorrecte. + +\end{itemize} + +\end{itemize} + +\clearpage +\subsubsection*{Manipulation 1} +On réalise à l'aide de Simulink l'échantillonnage et la reconstitution d'un signal de fréquence $f_0$ avec une période d'échantillonnage $T_e=1ms$ ($F_e=1kHz$). + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.35]{m1_f0100.png} + +\includegraphics[scale=0.35]{m1_f01200.png} + +\includegraphics[scale=0.35]{m1_f04010.png} +\caption{$s(t)$ et $e(t)$ pour $f_0$ = 100Hz, 1,2kHz et 4,01kHz} +\end{figure} + +\begin{itemize} +\item Lorsque $f_0 < F_e/2$, le signal est correctement reconstitué. +\item Lorsque $f_0 = 1,2F_e=1,2kHz$, comme prévu dans la préparation, on ne récupère qu'un signal avec une fréquence $f_0-F_e = 200Hz$. +\item Lorsque $f_0 \approx kF_e$ (ici $k=4$), alors on récupère de même un signal à la fréquence $f_0-kF_e$ (ici 4010-4000 = 10 Hz): c'est ce qu'on appelle un phénomène de battement. +\end{itemize} + +\subsubsection*{Préparation 2} +Le système comporte dans la chaîne directe le système analogique de fonction de transfert $H(p) = \frac{1}{1+2m_{BO}\frac{p}{\omega_{0BO}}+\frac{p^2}{\omega_{0BO}^2}}$ et un correcteur $C(p) = K_p$. +\begin{itemize} +\item L'expression de la fonction de transfert en boucle fermée s'écrit donc : +\begin{align*} +H_{BF}(p) & = \frac{H(p)C(p)}{1+H(p)C(p)} \\ +& = \frac{K_p}{(1+2m_{BO}\frac{p}{\omega_{0BO}}+\frac{p^2}{\omega_{0BO}^2}) + K_p} \\ +H_{BF}(p) & = \frac{\frac{K_p}{K_p+1}}{1+2\frac{m_{BO}}{K_p+1}\frac{p}{\omega_{0BO}}+\frac{p^2}{(K_p+1)\omega_{0BO}^2}} +\end{align*} +\item Avec $K_p=20$, on a alors +\begin{align*} +H_{BF}(p) & = \frac{0.95}{1+3,81.10^{-4}p+3,04.10^{-6}p^2} +\intertext{On a donc :} +\frac{1}{\omega_{0BF}^2} & = 3,04.10^{-6} \et +\frac{2m_{BF}}{\omega_{0BF}} = 3,81.10^{-4} +\intertext{On en déduit les paramètres en boucle fermée :} +\omega_{0BF} & = 574 rad.s^{-1} \\ +m_{BF} & = 0,11 +\end{align*} + +À l'aide de l'abaque donnant le temps de réponse réduit $t_r\omega_0$ en fonction de l'amortissement $m$, on en déduit $t_r\omega_0 = 30$ rad donc \[t_r = 52,3ms\] + +L'abaque des dépassements indiciels permet d'obtenir la valeur du premier dépassement +\[D_1 = 0.7\] + +\item Le diagramme de Bode est donné en annexe 1. On assure une marge de phase de $45^o$ si on a pour une pulsation donnée une phase de $-135^o$ et un gain nul. Pour le système non corrigé, on a un gain de $-7,16dB$ pour une phase de $-135^o$. Multiplier par un gain $K_{45}=10^{7,16/20}=2,28$ permet de translater la courbe de gain sans agir sur la phase, et on a alors bien la marge de phase voulue. +\end{itemize} + +\clearpage +\subsubsection*{Manipulation 2} +\paragraph{Simulation sur Simulink} On réalise l'asservissement analogique et on prend la réponse indicielle. + +Pour $K_p = 20$ on obtient la réponse indicielle donnée en annexe 5. La valeur finale est de 0.95, l'amplitude du premier dépassement est de 1.6. On a donc $D_1=\frac{1.6-0.95}{0.95}=0.68$. Le temps de réponse à $5\%$ est d'environ $50ms$. + +Pour $K_p = K_{45} = 2,28$, on se propose de relever le gain et la phase à $\omega=202rad/s$ (voir annexe 6).On a bien un gain unitaire et une phase de $-135^o$ donc le gain $K_{45}$ assure bien une marge de phase de $45^o$. + +\paragraph{Montage sur la maquette} +On réalise le montage sur la maquette. + +Pour $K_p=20=R_1/R_2$, on utilise $R_1=12K\Omega$ et $R_2=600\Omega$ en mettant deux résistances de $1,2k\Omega$ en parallèle. +On obtient la réponse indicielle suivante : +\begin{figure}[h!] +\centering +\includegraphics[scale=0.5]{m2Kp20.png} +\caption{Réponse indicielle, $K_p=20$} +\end{figure} + +Les curseurs nous permettent d'avoir un temps de réponse à $5\%$ de $52ms$ et un premier dépassement de $70\%$. + +\clearpage + +Pour $K_p=K_{45}$, on obtient la réponse suivante à $\omega = 202rad/s$ ($f=32.2Hz$) : + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.5]{m2K45w0dB.png} +\caption{$s(t)$ et $e(t)$ en boucle ouverte, $K=K_{45}$, $\omega = 202rad/s$} +\end{figure} + +On a bien un gain unitaire et un déphasage de $-135^o$, donc le gain $K_{45}$ assure bien une marge de phase de $45^o$. + +\bigskip + +\paragraph{Conclusion partie 2} Les résultats donnés par la simulation sont conformes à ceux obtenus par expérimentation sur la plaquette. La simulation Simulink représente bien le système réel implanté sur la plaquette. + +\clearpage + +\subsection*{Correcteur analogique numérisé} +\subsubsection*{Préparation 3} +On modélise l'action de l'échantillonneur-bloqueur par un retard pur de fonction de transfert \[B(p) = \exp(-T_ep/2)\] + +Ainsi, le gain de la fonction de transfert en boucle ouverte n'est pas affecté car $|\exp(-T_ep/2)| = 1$, et la phase est modifiée de $arg(\exp(-T_ep/2)) = -T_e/2 \omega$. + +\begin{figure}[h!] +\includegraphics[scale=0.4]{Bode2.png} +\caption{Diagrammes de Bode du système analogique et échantillonné} +\end{figure} + +\begin{itemize} +\item 1er cas : $T_e = 1ms$ (voir annexe 2)\\ + +Le gain correspondant à la phase $-135^o$ est de $-5.7dB$ pour $\omega=190rad/s$, donc on a $K_{45} = 10^{5.7/20} = 1.93$. + +La limite d'instabilité est atteinte lorsqu'on a simultanément une phase de $-180^o$ et un gain de $0dB$. On a ici un gain de $-23.9dB$ pour une phase de $-180^o$, donc on a à la limite d'instabilité $K_{lim}=10^{23.9/20}=15.7$. + + +\clearpage + +\item 2ème cas : $T_e = 5ms$ (voir annexe 3)\\ + +Le gain correspondant à la phase $-135^o$ est de $-2.5dB$ pour $\omega=150rad/s$, donc on a $K_{45} = 10^{2.5/20} = 1.33$. + +La limite d'instabilité est atteinte lorsqu'on a simultanément une phase de $-180^o$ et un gain de $0dB$. On a ici un gain de $-10,5dB$ pour une phase de $-180^o$, donc on a à la limite d'instabilité $K_{lim}=3,35$. + +\end{itemize} + +\clearpage + +\subsubsection*{Manipulation 3} + +On se propose dans les deux cas ($T_e=1ms$, $T_e=5ms$) de vérifier que le gain $K_{45}$ assure la marge de phase à $45^o$ et que le gain $K_{lim}$ est bien la limite de stabilité. + +\bigskip +\textbf{1er cas :} $T_e = 1ms$ + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.8]{m3retardpurTe1Kp45.png} +\caption{$e(t)$ et $s(t)$ en boucle ouverte, $K_p=K_{45}$, $\omega=190rad/s$} +\end{figure} + +Le gain $K_{45}$ assure bien une marge de phase de $-135^o$. + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.8]{m3retardpurTe1.png} +\caption{Réponse indicielle, $K=K_{lim}$} +\end{figure} + +Le gain $K_{lim}$ assure bien la limite d'instabilité. + +On se propose de comparer la simulation du système avec le correcteur numérique et avec sa modélisation par un retard pur. + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.8]{m3compTe1.png} +\caption{Réponse indicielle simulée pour $K=K_{45}$, avec correcteur numérique (--) et modélisation par un retard pur (+)} +\end{figure} + +La modélisation par un retard pur est très proche du comportement réel du correcteur. + +En faisant le montage avec la plaquette analogique et le correcteur numérique, on retrouve une réponse semblable. + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.5]{m3maquette.png} +\caption{Réponse indicielle avec la maquette et le correcteur numérique pour $K=K_{45}$} +\end{figure} + +\clearpage +\textbf{2ème cas :} $T_e = 5ms$ + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.8]{m3retardpurTe5Kp45.png} +\caption{$e(t)$ et $s(t)$ en boucle ouverte, $K_p=K_{45}$, $\omega=150rad/s$} +\end{figure} + +Le gain $K_{45}$ assure bien une marge de phase de $-135^o$. + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.8]{m3retardpurTe5.png} +\caption{Réponse indicielle, $K=K_{lim}$} +\end{figure} + +Le gain $K_{lim}$ assure bien la limite d'instabilité. + +\clearpage +\subsection*{Synthèse d'un correcteur PI numérisé} +\subsubsection*{Préparation 4} + +Le diagramme de Bode asymptotique du correcteur $C(p) = K_p(1+\frac{1}{T_ip})$ est le suivant : + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\draw [>=latex,->] (0,0) -- (0,3) node[left]{$\phi(^o)$} ; +\draw [>=latex,->] (-1,2) -- (6,2) node[right]{$\omega$}; + +\draw [red] (0,1) node[left]{$-90$} -- (3,1) -- (3,2) -- (6,2); + +\draw [>=latex,->] (0,4) -- (0,7) node[left]{$G_{dB}$} ; +\draw [>=latex,->] (-1,5) -- (6,5) node[right]{$\omega$}; + +\draw [red] (0,6.5) -- (3,5.5) -- (6,5.5); +\draw [red,dashed] (0,5.5) node[left]{$K_p$} -- (3,5.5) -- (3,2) node[left]{$\frac{1}{T_i}$}; + +\end{tikzpicture} +\end{figure} + +Pour obtenir une marge de phase de $-45^o$, il faut utiliser $K_p$ pour ajuster la courbe de gain. Pour cela, on se place dans le domaine où le gain du correcteur est égal à $K_p$ et où la phase est nulle. + +Le gain $K_p$ utilisé sera celui calculé précédemment, $K_{45} = 2.28$. +On avait pour le système non corrigé une phase de $-135dB$ pour une pulsation de 202 rad/s. On va donc placer $\frac{1}{T_i}=\frac{202}{10}$ donc $T_i=49.5ms$. + +\end{document} \ No newline at end of file diff --git a/421-Controle_processus/TP1/bode.odt b/421-Controle_processus/TP1/bode.odt new file mode 100644 index 0000000000000000000000000000000000000000..99c264d498554537ced5e765295cf397c0b64e88 GIT binary patch literal 206897 zcmbrlWl&sA)Gj*M;O@aCxCM6@++9NO;6Z~E+&y@J1R31j-QC^Y-QDiI-#PczIlu3Z zJ=L{qYFqbOy&hRA@-VPC00062uv(%lY0%G(LJI%@{yQM20Jc`PX3icCX2uQ|Q^t_sW?;hM;!2VsT@1i^du>A( zUWPwnsCsR8i)W(WHdVr5AW*~xiHgM!#oDsJr0z1dve8P3+#V z;YJoILU}||$B=%ZXdem7?i7YX3Zq8|2m**83d(1i@Y4n-YKAxGVbx?CVLO@`>rjFW z*bvo~rtzi&D1)XD0j{4OO=^}Nl|HQ}g);#u@7dzQ5>=WW#0N&J*IDme)-O_-u6!V0 ziz3}kiw1wXAcZ*rlN`~xT7nu2u4tl%NXEQPZ*U0LM?Cdwm7yAiQ*}f~nXM=u8^Ik4 zIU26M&J)ok>xOn9V1y)X@hdb_>`Xu?3asEjaSp9NOQfQxYUOi&z2mOLOI_>iwYiy9 zpHGN7c>+oU!ar}*e<4N!Q!l|%q!ZX&L}+Pi@0oT(Bllavm;xVaPjxvhwpdAh4%6=N zdp%>McVr39&%nN6y?d^1Zf?}m**!GWGxRNETh2(?V|8|Cf8PWQpI&NK7=JMSsrbgR 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