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zTC8tzGbgfwRajxd)=Gx^(c0W{A!&}z(H$Wyw)c;mBEVeK&MrFMPD@UyeV3^aI}Hrn zUnY23?hD#5r2JN6f*0X9Cc7?{mD8rAu$v`3)5SW1B4aS#`tOv_4HIv{fi<-Pu>y5m zkL^>KuP~O|Eb|7DWOO&u2|m6eam=@|J1Bv7h0NhYi?tQ|YmZ<`E{oPOxp8i*v^FWa z=f3l}L9hn(M!x5uMJV$bB&Qa#>}fCIT3&b>Wo65DOWkI1cPCQ~`0$2AhL|Y6Lxm4+ zqV3XdmuKwY{^KcH0$ tel que $\forall t > 0$, $x(t+T) = x(t)$.\\ +(On exclut cependant le cas $x(t)$ = constante). + + +\paragraph{Cycle limite stable}:\\ +Pour toutes les conditions initiales appartenant au voisinage du cycle limite,\\ +\[\exists t_0 > 0 \text{ et }T > 0 \text{ tel que } \forall t>t_0, \quad x(t+T) = x(t)\] +i.e. toute trajectoire dans un voisinage du cycle limite converge dans un temps fini vers le cycle limite. + +\paragraph{Cycle limite instable}:\\ +Toutes les trajectoires divergent du cycle limite.\\ +Pour toutes les CI n'appartenant pas au cycle limite, $ \exists t > 0 \text{ tel que} x(t) \notin \text{cycle limite} $. + +\paragraph{Cycle semi-stable}:\\ + Une partie des trajectoires converge et d'autres divergent du cycle limite. + +\section*{Théorème de Bendixon} + +\begin{thm} +Soit le système du second ordre $\dot{x}=f(x)$ avec $f$ le champ de vecteurs tel que $f:D\rightarrow\R^2$ avec $D$ un connexe (d'un seul tenant, non formé de la réunion d'ensemble disjoint) de $\R^2$ ne contenant pas de point d'équilibre. +Si: +\begin{itemize} +\item $\exists x \in D$ tel que $\div f(x) \neq 0$ +\item $\div f$ ne change pas de signe dans $D$ +\end{itemize} +Alors $\dot{x}=f(x)$ n'a pas de cycle limite inclus dans $D$. +\end{thm} + +\begin{proof} +Par l'absurde, soit $\Gamma = \{x\in D, x(t), 0 \leq t \leq T\}$ est un cycle limite. + +$\forall x \in \Gamma$, $f(x)$ est tangent à $\Gamma$ tel que $f(x).n(x)=0$ où $n(x)$ est le vecteur normal de $\Gamma$ en $x$. + +Suivant le théorème de Green, +\[ \oint_{\Gamma} f(x)n(x)dx = \iint_S \div f(x)dS \text{ donc } \iint_S \div f(x)dS = 0\] + +Si $\exists x \in D$ tel que $\div f(x) \neq 0$ et que $\div f$ ne change pas de signe dans $D$ (donc a fortiori dans $S\subset D$), on déduit de la continuité de l'opérateur $\div f$ dans $D$ que $\iint_S \div f(x)dS \neq 0$ : contradictoire. + +Ainsi, $D$ ne contient pas de cycle limite. +\end{proof} + +\begin{example} +Soit le système NL du 2nd ordre $\ddot{x}(t) + \alpha \dot{x}(t) + g(x(t)) = 0$, avec $x(0) = x_0$ et $\dot{x}(0) = \dot{x}_0$ où $\alpha \neq 0$ et $g:\R \rightarrow \R$ continue avec $g(0)=0$. \\ + +Représentation d'état : + +\[ + \begin{cases} +\dot{x}_1(t) & = x_2(t) = f_1(x)\\ + \dot{x}_2(t) & = - \alpha x_2(t) - g(x_1(t)) = f_2(x) +\end{cases} +\text{ avec } x_1(t) = x(t) \text{ et }x_2(t) = \dot{x}(t) \] + +Calculons $\div f = \derivp[f_1]{x_1} + \derivp[f_2]{x_2} = -\alpha$. + +$\div f \neq 0$ et ne change pas de signe donc ce système ne comporte pas de cycle limite $(D=\R^2)$. +\end{example} + +\section*{Théorème de Poincaré-Bendixon} + +\begin{thm} +Soient le système du 2nd ordre $\dot{x}=f(x)$ et $O_{x_0}^+$ une trajectoire positive, i.e $O_{x_0}^+ = \{ x \in D, x = S(t,x_0), t \geq 0\}$ où $S(.,x) : \R \rightarrow D$ définit une solution de $\dot{x}=f(x)$ pour une trajectoire passant par $x$, avec un ensemble limite $\omega(x_0)$ i.e. $\omega(x_0) = \bigcap_{t \geq 0} \overline{O_{x_0}^+}$ \footnote{adhérence = plus petit fermé contenant l'ensemble}\\ + +Si $\omega(x_0)$ est compact et ne contient pas de point d'équilibre, alors la limite ne peut être qu'un cycle limite.\\ +\end{thm} + +Interprétation : + +Dans le cas du 2nd ordre, si on a une convergence des trajectoires vers un compact (fermé borné de $\R^2$) qui ne contient pas de point d'équilibre, alors la limite ne peut être qu'un cycle limite.\\ +\paragraph{Examples du poly page 4} Système hybride = commutation entre 2 systèmes linéaires\\ + +Example 1 : +\begin{align*} +\dot{x} & = + \begin{bmatrix} +-1 & 10 \\-100 & -1 x = A_1x +\end{bmatrix}\\ +\dot{x} & = + \begin{bmatrix} +-1 & 100 \\ -10 & -1 x = A_2x +\end{bmatrix} +\quad \text{v.p. } \lambda_{1,2} = -1 \pm j31,62 +\end{align*} + +Les deux systèmes sont stables + + + +Stabilité locale mais le système est instable globalement.\\ + +Important : l'analyse faite par linéarisation donne uniquement une information sur la stabilité locale et non globale.\\ + +Exemple 2 : +\begin{align*} +\dot{x} & = + \begin{bmatrix} +1 &- 10\\100 & 1 x +\end{bmatrix} += A_1x \\ +\dot{x} & = + \begin{bmatrix} +1 & -100\\10 & 1 +\end{bmatrix} +x = A_2x \quad \text{v.p. } \lambda_{1,2} = -1 \pm j31,62 +\end{align*} + +Les deux systèmes sont instables. + +En choisissant bien la permutation, on rend le système global stable. + +\paragraph{Conclusion} l'analyse de la stabilité par linéarisation ne donne pas une CNS de stabilité des systèmes non linéaires (point d'équilibre), d'où l'importance de définir un autre moyen d'analyse. \\ + +\begin{rem} +Il existe d'autres méthodes pour tracer les trajectoires dans le plan de phase. +\end{rem} + +\begin{example}[Élimination du temps] +\begin{multicols}{2} +\noindent Méthode explicite : +\[ + \begin{cases} +x_1(t) & = x_0 \cos t + \dot{x}_0 \sin t\\x_2(t) & = -x_0 \sin t + x_0 \cos t +\end{cases} +\] + +\[x_1^2(t) + x_2^2(t) = x_0^2 + \dot{x}_0^2 \] +On a éliminé le temps mais c'est assez \emph{spicifique} à la représentation d'état. + +\noindent Méthode implicite : +\[ \dot{x} = + \begin{bmatrix} +0 & 1 \\ 1 & 0 +\end{bmatrix} +x \text{ donc } + \begin{cases} + \dd{x_1}{t} & = x_2\\ \dd{x_2}{t} & = -x_1 +\end{cases} +\] +\[dt = \frac{dx_1}{x_2} = -\frac{dx_2}{x_1}\] +\[x_1dx_1 = -x_2dx_2 \text{ donc } x_1^2 + x_2^2 = x_{20}^2 + x_{10}^2\] +\end{multicols} +\end{example} +\end{document} diff --git a/424-Systeme_Non_Lineaires/Cours/chap3.tex b/424-Systeme_Non_Lineaires/Cours/chap3.tex new file mode 100644 index 0000000..35633ce --- /dev/null +++ b/424-Systeme_Non_Lineaires/Cours/chap3.tex @@ -0,0 +1,295 @@ +\documentclass[main.tex]{subfiles} + +\begin{document} +\section{Hypothèses} +\begin{itemize} +\item la non-linéarité est statique et n'évolue pas dans le temps. On peut la séparer de la dynamique du système. Par exemple, la saturation (ou la zone morte) est une non-linéarité statique. +\item la partie dynamique (linéaire) est un filtre passe-bas \emph{suffisamment efficace} pour négliger les harmoniques d'ordre supérieur à 1. Plus précisément, l'ordre relatif du filtre doit être supérieur strict à 1. +\end{itemize} + +\section{Schéma-blocs} +\[ x \longrightarrow \boxed{\text{Non-linéarité}} \longrightarrow y \longrightarrow \boxed{H(p)} \longrightarrow z \] +La fonction de transfert $H(p)$ (fraction rationnelle) correspond à un filtre passe-bas de degré relatif $\geq 2$.\\ + +On prend $x=X\sin \omega t$. Dans le cas linéaire, seule la valeur de $\omega$ influe sur le tracé de la diagramme de Bode du système. Dans le cas non-linéaire, on a plusieurs tracés de réponses fréquentielles. Par exemple, avec une saturation, on obtient des réponses fréquentielles qui dépendent de l'amplitude d'entrée de $X$ dès qu'elle devient trop élevée. + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.4]{2/424-1.png} +\end{figure} + +Puisque $H(p)$ rejette les harmoniques d'ordre supérieur à 1, on peut donc décomposer \[y(t)=P \sin \omega t + Q \cos \omega t\] + +Dans le cas d'une NL symétrique, on a +\begin{align*} +P& =\frac{2}{T} \int_{[T]} y(t) \sin \omega t dt\\ +Q& =\frac{2}{T} \int_{[T]} y(t) \cos \omega t dt \quad \text{ avec } \omega T = 2\pi +\end{align*} + +\begin{rem} +Si la NL est non-symétrique, $y(t) = Y+P\sin \omega t + Q \cos \omega t$ avec $Y=\frac{1}{T}\int_{[T]} y(t) dt$. La composante continue $Y$ peut être négligée pour l'analyse de stabilité et modélisée par une perturbation constante à l'entrée de $H(p)$. +\end{rem} + +On définit le gain complexe équivalent : +\[ N(X) = \frac{P+jQ}{X} \text{ qu'on note } N(x) = N_P(X) + jN_Q(X) \] +\begin{itemize} +\item $N_P(X)=\frac{P}{X}$ est la gain en phase, +\item $N_Q(X)=\frac{Q}{X}$ est la gain en quadrature. +\end{itemize} + +\begin{rem} +\begin{itemize} +\item À la différence du système linéaire, pour une même pulsation, on a plusieurs réponses fréquentielles qui dépendent de l'amplitude de l'entrée $X$. L'analyse de stabilité doit donc se faire par rapport à tous les tracés. + +% Inclure le nyquist du génie + +\item Les manipulations de schéma-blocs doivent satisfaire les règles connues (principe de superposition) et s'assurer que le signal en amont du bloc NL est le même, et en aval, qu'il est suffisamment filtré pour ne garer que le 1er harmonique. + +\begin{example} + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\sbEntree{E} + +\sbComp[3]{comp}{E} +\sbRelier[$e$]{E}{comp} + +\sbBloc[2]{C}{$C(p)$}{comp} +\sbRelier{comp}{C} + +\sbBloc[2]{NL}{Non-linéarité}{C} +\sbRelier[$x$]{C}{NL} + +\sbBloc[2]{sys}{$H(p)$}{NL} +\sbRelier{NL}{sys} + +\sbSortie[2]{S}{sys} +\sbRelier{sys}{S} + +\sbRenvoi{sys-S}{comp}{} + +\end{tikzpicture} + +\vspace{5mm} +équivalent à +\vspace{5mm} + +\begin{tikzpicture} +\sbEntree{E} + +\sbBloc[3]{C}{$C(p)$}{E} +\sbRelier[$e$]{E}{C} + +\sbComp[4]{comp}{C} +\sbRelier{C}{comp} + +\sbBloc[2]{NL}{Non-linéarité}{comp} +\sbRelier[$x$]{comp}{NL} + +\sbBloc[2]{sys}{$H(p)$}{NL} +\sbRelier{NL}{sys} + +\sbSortie[2]{S}{sys} +\sbRelier{sys}{S} + +\sbDecaleNoeudy[4]{S}{R} +\sbBlocr[8]{Cr}{$C(p)$}{R} +\sbRelieryx{sys-S}{Cr} +\sbRelierxy{Cr}{comp} + +\end{tikzpicture} + +\vspace{5mm} +non équivalent à +\vspace{5mm} + +\begin{tikzpicture} +\sbEntree{E} + +\sbComp[3]{comp}{E} +\sbRelier[$e$]{E}{comp} + +\sbBloc[4]{NL}{Non-linéarité}{comp} +\sbRelier[$\hat{x}\neq x$]{comp}{NL} + +\sbBloc[2]{sys}{$H(p)C(p)$}{NL} +\sbRelier{NL}{sys} + +\sbSortie[2]{S}{sys} +\sbRelier{sys}{S} + +\sbRenvoi{sys-S}{comp}{} + +\end{tikzpicture} +\caption{Transformations de schéma-blocs} +\end{figure} + +\end{example} +\end{itemize} +\end{rem} + +\newpage +\section{Analyse de la stabilité.} + +Système NL bouclé à retour unitaire + +\begin{figure}[h!] +\centering +\begin{tikzpicture} +\sbEntree{E} + +\sbComp[4]{comp}{E} +\sbRelier[$e$]{E}{comp} + +\sbBloc[4]{NL}{$N(X)$}{comp} +\sbRelier[$x$]{comp}{NL} + +\sbBloc[4]{sys}{$T_{BO}(p)$}{NL} +\sbRelier{NL}{sys} + +\sbSortie[4]{S}{sys} +\sbRelier{sys}{S} + +\sbRenvoi{sys-S}{comp}{} + +\end{tikzpicture} +\end{figure} + +Dans l'analyse harmonique, la NL est modélisée par $N(X)$. Ainsi, il faut trouver l'expression de $N(X)$ en fonction de la NL : + +\begin{example}[Saturation] + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.4]{2/424-4.png} +\end{figure} + +Calcul de $N(X)$ : + +Pour $0 \leq t \leq t_1$ : $y(t) = X\sin \omega t$ + +$t_1 \leq t \leq \frac{\pi}{\omega}-t_1$ : $y(t) = X_m = X\sin \omega t_1$ + +\begin{align*} +P & = \frac{4\omega}{\pi} \int_0^{\frac{\pi}{2\omega}} y(t) \sin \omega t dt \\ +& = \frac{4\omega}{\pi} [ \int_0^{t_1} X \sin^2 \omega t dt + \int_{t_1}^{\frac{\pi}{2\omega}} X \sin \omega t_1 \sin \omega t dt ] \\ +& = \frac{2X}{\pi}[ \omega t_1 + \frac{\sin 2\omega t_1}{2} ] \\ +\intertext{ $t_1=\arcsin(\frac{X_m}{X})$ et $Q=0$} +\intertext{Ainsi} +N(x) & = + \begin{cases} + 1 & \si X << X_m\\ + \frac{2}{\pi}[\arcsin\frac{X_m}{X}+\frac{X_m}{X}\sqrt{1-\frac{X_m^2}{X^2}}] & \si X > X_m + \end{cases} +\end{align*} +Le dénominateur de la BF, $1+N(X)T_{BO}(p)$, donne la limite de stabilité : $T_{BO}(j\omega) = - \frac{1}{N(X)}$. + +On a donc pour notre exemple de saturation + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.2]{2/424-5.png} +\end{figure} + +\end{example} + +Ainsi dans le cas NL, on remplace le point critique $-1$ par le lieu critique $\frac{-1}{N(X)}$. Par conséquent, l'analyse de stabilité est réalisée par rapport à $\frac{-1}{N(X)}$. + +Dans le cas où le tracé de Nyquist ne présente pas d'intersection avec le lieu critique, on applique le critère de Nyquist pour la stabilité ou celui du revers sur la FT, qui est alors stable, strictement propre et à déphasage minimal. + +Si on a une ou plusieurs intersections, on a un régime auto-oscillant (cycle limite). + +\section{Étude de la stabilité du cycle limite} + +Soit $(X_0,\omega_0)$ solution de $T_{B0}(j\omega_0)=-\frac{1}{N(X_0)}$.\\\ + +\subsection*{Critère analytique} +\begin{multicols}{2} +On décompose \[1+T_{B0}(j\omega)N(X)=R(\omega,X)+jI(\omega,X)\] + +Ainsi, on a \[R(\omega_0,X_0)=0 \text{ et }I(\omega_0,X_0)=0\] +\end{multicols} + +Par conséquent, l'entrée de la NL $x(t)=X_0e^{j\omega_0t}$\\ + +À $t_0$ appliquons une perturbation : +\[X_1 = X_0 + \delta X \text{ et }\omega_1 = \omega_0+\delta \omega \quad \text{ avec } |\frac{\delta X}{X_0}|<<1 \text{ et }|\frac{\delta \omega}{\omega_0}|<<1 \] + +$x(t)$ n'est plus périodique et présente ainsi un amortissement $m>0$ (stable) ou $<0$ (instable). + +\[ x(t) = (X_0 + \delta X) e^{-mt} e^{j(\omega_0 + \delta \omega) t} =(X_0 + \delta X) e^{j(\omega_0 + \delta \omega + jm) t} \] + +Ainsi la perturbation nous donne un régime auto-oscillant avec une amplitude $X_0+\delta X$ et une pulsation complexe $\omega_0 + \delta \omega + jm$. + +\[ R(\omega_0+\delta \omega + jm, X_0 + \delta X) + jI(\omega_0 + \delta \omega + jm,X_0 + \delta X) = 0 \] + +\newcommand{\zero}{(\omega_0,X_0)} +On applique un DL du 1er ordre autour de $\zero$ : +\[ \left(\derivp[R]{X}|_{\zero} + j \derivp[I]{X}|_{\zero}\right) \delta X + \left(\derivp[R]{\omega}|_{\zero} + j \derivp[I]{\omega}|_{\zero}\right)(\delta \omega + jm)\approx 0 \] +i.e. en notant $\derivp[]{X}|_{\zero}=\derivp[]{X}|_0$ +\begin{align*} +\derivp[R]{\omega}|_0 .\delta \omega + \derivp[R]{X}|_0 .\delta X - \derivp[I]{\omega}|_0 .m & = 0 \\ +\text{ et }\quad \derivp[I]{\omega}|_0 .\delta \omega + \derivp[I]{X}|_0 .\delta X + \derivp[R]{\omega}|_0 .m & = 0 +\end{align*} + +Élimination de $\delta \omega$ : +\[ \left( (\derivp[R]{\omega}|_0)^2 + (\derivp[I]{\omega}|_0)^2 \right) m = \left( \derivp[R]{X}|_0.\derivp[I]{\omega}|_0 - \derivp[R]{\omega}|_0.\derivp[I]{X}|_0 \right) \delta X \] + +\newpage +\noindent Différents types de perturbation + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.4]{2/424-61.png} +\end{figure} +$m > 0$ et $\delta X > 0$ : CL est stable + +$m < 0$ et $\delta X > 0$ : CL est instable + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.4]{2/424-62.png} +\end{figure} +$\delta X < 0$ et $m < 0$ : CL est stable + +$\delta X < 0$ et $m > 0$ : CL est instable + +Donc le cycle limite est stable si et seulement si $\delta X . m >0$\\ + +Pour que $\delta X . m >0$ : +\[\boxed{ \derivp[R]{X}|_0.\derivp[I]{\omega}|_0 - \derivp[R]{\omega}|_0.\derivp[I]{X}|_0 > 0 }\] +\begin{center} +\emph{Condition de stabilité du cycle limite dans le plan de Nyquist} +\end{center} + +On pose $T_{B0}(j\omega) = U(\omega) + jV(\omega)$ et $-\frac{1}{N(X)} = L(X) + jM(X)$ + +On a un cycle limite si +\[ T_{B0}(j\omega_0) = -\frac{1}{N(x)} \quad \Rightarrow \quad + \begin{cases} +R(\omega,X) & = U(\omega) - L(X)\\I(\omega,X) & = V(\omega) - M(X) +\end{cases} +\] + +d'où d'après la condition de stabilité du cycle limite : +\[\boxed{-\derivp[L]{X}|_0.\derivp[V]{\omega}|_0 + \derivp[U]{\omega}|_0.\derivp[M]{X}|_0 > 0}\] + +\subsection*{Détermination graphique} +Si on se place dans $\R^3$, on a 2 vecteurs : $\vect{U\\V\\0}$ et $\vect{L\\M\\0}$ qui décrivent respectivement $T_{BO}$ et $-\frac{1}{N}$. + +Les tangentes aux courbes $T_{BO}$ et $-\frac{1}{N}$ sont colinéaires aux vecteurs $\vec{v_T} = \derivp[]{\omega}\vect{U\\V\\0}$ et $\vec{u_N}=\derivp[]{X}\vect{L\\M\\0}$. + +\[\vec{v_T}\wedge\vec{u_N} = \vect{0\\0\\-\derivp[L]{X}.\derivp[V]{\omega} + \derivp[U]{\omega}.\derivp[M]{X}}\] + +Ainsi, la condition $-\derivp[L]{X}.\derivp[V]{\omega} + \derivp[U]{\omega}.\derivp[M]{X}>0 \Rightarrow (\vec{v_T},\vec{u_N})$ dans le sens direct. + +\begin{thm}[Critère de Loeb] +Le cycle limite est stable si l'intersection de $T_{BO}(j\omega)$ et de $-\frac{1}{N(X)}$ est telle qu'en parcourant le lieu de Nyquist $T_{BO}(j\omega)$ dans le sens des $\omega$ croissants, on laisse à gauche la direction des $X$ croissant sur le lieu critique. +\end{thm} + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.4]{2/424-7.png} +\end{figure} +\end{document} diff --git a/424-Systeme_Non_Lineaires/Cours/chap4.tex b/424-Systeme_Non_Lineaires/Cours/chap4.tex new file mode 100644 index 0000000..6658219 --- /dev/null +++ b/424-Systeme_Non_Lineaires/Cours/chap4.tex @@ -0,0 +1,746 @@ +\documentclass[main.tex]{subfiles} +\newcommand{\D}{\mathcal{D}} +\newcommand{\Kc}{\mathcal{K}} +\newcommand{\Lc}{\mathcal{L}} +\begin{document} + +\section{Trajectoire} + +\begin{rem} +Dans le cas linéaire, la trajectoire est la solution au système $\dot{x}=Ax$ avec $x(0)=x_0$. Cette solution est unique. Qu'en est-il en non-linéaire? +\end{rem} + +\begin{defin} +Un système dynamique sur $\D \subset \R^n$, où $n$ est la dimension du système, est un triplet $(\D,\R,s)$ où $s:\R \times \D \rightarrow \D$ tel que les axiomes suivants sont vérifiés : +\begin{enumerate} +\item Continuité : $s(.,.)$ est continue sur $\R \times \D$ et $\forall t \in \R$, $s(.,x)$ est dérivable. +\item Consistance : $s(0,x_0)=x_0$, $\forall x_0\in \D$. +\item Propriété de groupe : $s(\tau, s(t,x_0)) = s(t+\tau,x_0)$, $\forall x_0\in \D$. +\end{enumerate} +\end{defin} + +\begin{rem} +\begin{itemize} +\item On dénote le système $(\D,\R,s)$ par $G$, où $s(.,.)$ est la trajectoire et $\D$ est l'espace de phase. +\item On dénote la trajectoire $s(t,.) : \D \rightarrow\D$ par $s_t(x_0)$ ou $s_t$. +\item Suivant l'axiome de consistance, $s_0(x_0)=x_0$ et suivant la propriété de groupe : +\[ (s_{\tau} \circ s_t)(x_0) = (s_t \circ s_{\tau})(x_0) = s_{t+\tau}(x_0) \] +Ainsi l'application inverse de $s_t$ est $s_{-t}$ où $s_t$ est un homéomorphisme (bijective, continue, inverse continue). + +En effet, montrons que $s_t$ est injective. + +Soit $y,z\in \D$ tels que $s_t(z)=s_t(y)$. +On a $z=s_0(z)=s(0,z)=s(t-t,z)=s(-t,s(t,z))=s(-t,s(t,y))=s(0,y)=y$ + +$s_t$ est surjective : $\forall z \in D, \exists y \in \D$ tel que $y=s(-t,z)$. + +Enfin, $s_t$ est continue sur $\R$ donc $s_{-t}$ est continue. +\end{itemize} +\end{rem} + +\begin{exemple} +Système linéaire causal de dimension $n$ ($n$ variables d'état) + +$s:[0,+\infty[ \times \R^n \rightarrow \R^n$ où $s(t,x)=e^{At}x$ où $A\in\R^n$ matrice d'évolution + +Ainsi $s_t(x) = e^{At}x$ où $s_t : +\begin{cases} +\R^n & \rightarrow \R\\x & \mapsto e^{At}x +\end{cases} +$ + +On a $(s_{\tau} \circ s_t) (x) = s_{\tau}(s_t(x)) = e^{A\tau}e^{At}x = s_{t+\tau}(x)$ +\end{exemple} + +\begin{prop} +Suivant l'axiome 1, le système $G$ peut être décrit par une équation différentielle sur $\D$. En particulier, la fonction $f:\D \rightarrow \R^n$ définie par $f(x) = \dd{s(t,x)}{t}|_{t=0}$. Ainsi, $f(x)$ est un champ de vecteur sur $\D$ où pour $x\in\D,f(x)\in\R^n$ correspond au vecteur tangent à la trajectoire en $t=0$. +\end{prop} + +\begin{exemple} +Système linéaire $f(x)=\dd{e^{At}x}{t}|_{t=0}=Ax$ +\end{exemple} + +\begin{rem} +Nous avons défini une trajectoire, mais à partir de $\dot{x}=f(x)$, est-elle unique ? +\end{rem} + +\section{Théorème du point fixe} + +\begin{thm} +Soient $X$ un espace de Banach de norme $||.||$, $S$ un fermé de $X$ et $T:S\rightarrow S$ une application contractante sur $S$, i.e. $\exists \rho \in [0,1[$ tel que $\forall (x,y) \in S^2, ||T(x)-T(y)|| \leq \rho ||x-y||$, + +\[ \text{ alors } \exists ! x^* \in S \text{ tel que } T(x^*)=x^*\] + +De plus, quelque soit la suite sur $S$ tel que $x_{n+1}=T(x_n)$, elle converge vers $ x^* $. +\end{thm} + +\begin{defin} +Une application $f:(X,d_x) \rightarrow (Y,d_y)$ est lipschitzienne si $\exists \alpha > 0$ tel que \[\forall x,y \in X, \quad d_y(f(x),f(y)) \leq \alpha d_x(x,y)\] +\end{defin} + +\begin{rem} +Une fonction lipschitzienne est uniformément continue. +\end{rem} + +\begin{thm} +Soient le système dynamique défini par :$\dot{x}(t)=f(x(t))$ et $x(t_0)=x_0, t \in \R (*)$. + +Si $f:D \rightarrow \R^n$ est lipschitzienne sur $D$ alors $\forall x_0 \in \D, \exists \tau \in ]t_0,t_1[$ tel que (*) a une unique solution $x:[t_0,\tau] \rightarrow \R^n$ +\end{thm} + +\begin{proof} +Soient $T(x) = x_0 + \int_t^{t_0}f(s)ds$, $t\in[t_0,\tau] = x(t)$ + +et on définit $S = \{ x(t) \text{ tel que } t\in [t_0,\tau], ||x-x_0|| \leq r \}$ + +Ainsi, $\forall x \in S$ +\begin{align*} +||T(x) - x_0|| & = ||\int_{t_0}^t f(s)ds || \\ +& = || \int_{t_0}^t (f(s)-f(t_0)+f(t_0))ds || \\ +& \leq \int_{t_0}^t ||f(s)-f(t_0)||s + \int_{t_0}^t ||f(x_0)||ds \\ +& \leq (\alpha r + C) ds \quad (f \text{ lipsch. et } ||s-x_0|| \leq r) \\ +& \leq (\alpha r + C)(t-t_0) \leq r +\end{align*} + +$\exists \tau \in ]t_0,t_1[$ tel que $(\tau - t_0) \leq \frac{r}{\alpha r + C}$ donc $T:S\rightarrow S$. + +\begin{align*} +\forall x,y \in S, \quad ||T(x)-T(y)|| & \leq \int_{t_0}^t || f(x(s))-f(y(s)) || ds \\ +& \leq \alpha \int_{t_0}^t || x(s) - y(s) || ds \\ +& \leq \alpha \max_{s\in [t_0,\tau]} ||x(s)-y(s)|| \int_{t_0}^t ds \\ +& \leq \alpha |||x(s)-y(s)||| (t-t_0) \quad \text{ avec } |||.|||=\max_{s\in [t_0,\tau]}(.) +\end{align*} + +On veut $\alpha (t-t_0) \leq \alpha (\tau - t_0) \leq \rho$ avec $\rho<1$ donc $|||T(x)-T(y)|| \leq \rho |||x-y|||$. + +Il suffit de choisir $\tau$ tel que $\tau - t_0 \leq \frac{\rho}{\alpha}$ + +$T:S \rightarrow S$ est contractante pour $\tau - t_0 \leq \min \{ \frac{r}{\alpha r + C}, \frac{\rho}{\alpha} \}$ + +(*) a une unique trajectoire. +\end{proof} + +\begin{exemple} +Soit le système $\dot{x}(t) = \sqrt{|x(t)|}$, $x(0)=0$, $t\geq 0$ + +Ce système a une infinité de solutions paramétrées par $T$ +\[ + \begin{cases} +x(t) = 0 & \si 0 \leq t \leq T\\ \frac{(t-T)^2}{4} & \si t > T +\end{cases} +\] + +$\forall x,y \in \R_+$, +\[ |\sqrt{x} - \sqrt{y}| = \frac{|x-y|}{\sqrt{x}+\sqrt{y}} \leq \alpha |x-y| \] +donc $\frac{1}{\sqrt{x}+\sqrt{y}} \leq \alpha$. + +Ainsi, si $x$ et $y$ sont proches de 0, on peut rendre la partie à gauche de l'inégalité arbitrairement grande. +\end{exemple} + +\section{Attracteur} + +\begin{defin} +Un ensemble $M \subset D$ est positivement invariant du système + +\[G:\dot{x}(t) = f(x(t)), x(0) = x_0, t\in \R \quad (*) \] + +si $s_t(M) \subseteq M$ pour $t\geq 0$ où $s_t(M) = \{ s_t(x), x\in M \}$.\\ + +Il est négativement invariant suivant la dynamique (*) si $s_t(M) \subseteq M$ pour $t<0$. Ainsi $M$ est un ensemble invariant suivant (*) si $s_t(M) \subseteq M, \quad \forall t \in \R$ +\end{defin} + +\begin{prop} +Si $M \subset D$ est un ensemble invariant suivant (*), alors $\overline{M}$ l'adhérence de $M$ est invariant. +\end{prop} + +\begin{proof} +Soit la suite $(x_n)_{n\in\N} \subset M$ tel que $x_n \rightarrow x$ avec $x\in \overline{M}$. + +Puisque $M$ est invariant, alors $(s_t(x_n))_{n\in\N} \subset M$. De plus, $s_t(x_n) \rightarrow s_t(x) \in \overline{M}$ car c'est un fermé. + +Ainsi, $\overline{M}$ est invariant suivant (*). +\end{proof} + +\begin{defin}[Attracteur] +Un ensemble invariant fermé $M \subset D$ est un attracteur du système (*), s'il existe un voisinage $N$ de $M$ tel que $\forall x \in N, \exists t \in \R$ tel que $s_t(x) \in M$ +\end{defin} + +\begin{rem} +Un cycle limite stable ou semi-stable est un attracteur. +\end{rem} + +\begin{exemple} +Soit le système : +\[ + \begin{cases} +\dot{x_1}(t) = -x_2(t) + x_1(t)(1-x_1^2(t) - x_2^2(t)) \\ \dot{x_2}(t) = x_1(t) + x_2(t)(1-x_1^2(t) - x_2^2(t)) +\end{cases} +\] + +En utilisant les coordonnées polaires, on trouve l'attracteur de $M$.\\ + +On a en effet +$r = \sqrt{x_1^2 + x_2^2}$ et $ \theta = \arctan\frac{x_2}{x_1}$ + +donc +$\dot{r} = \derivp[r]{x_1} \dot{x_1} + \derivp[r]{x_2}\dot{x_2} = r(1-r^2)$ et $\dot{\theta} = \derivp[\theta]{x_1}\dot{x_1} + \derivp[\theta]{x_2}\dot{x_2} = 1$\\ + +Ainsi, + +$r>1 \quad \dot{r}<0 \Rightarrow r \rightarrow 1$ + +$r<1 \quad \dot{r}>0 \Rightarrow r \rightarrow 1$ + +$r=1$ un fermé $\Rightarrow$ Attracteur où $\forall (x_1,x_2) \in \R^2 / \{(0,0)\}$ car $x_1=x_2=0$ est un point d'équilibre, les trajectoires convergent vers le cercle unité. Suivant le théorème de Poincaré-Bendixon le cercle unité est un cycle limite, car c'est un compact et ne contient pas de point d'équilibre. +\end{exemple} + +\section{Types de stabilité en non linéaire} + +\paragraph{Stabilité suivant Lagrange} Le concept est basé sur l'énergie potentielle $V$. Puisque lesp oints d'équilibre du système correspondent aux points tels que $\derivp[V]{q}=0$ avec $q$ les coordonnées généralisées du mouvement, alors un point d'équilibre est stable suivant Lagrange si $\derivpp[V]{q} > 0$ + +%\img{0.3}{3/1.png} %HALLELUJAH ! + +Suivant Lagrange, un point d'équilibre est stable si de faibles perturbations induisent de faibles variations de la solution (trajectoire). + +\begin{rem} +La notion de stabilité en non linéaire concerne les points d'équilibre et non le système. Dirichlet a formalisé la stabilité au sens de Lagrange avec les trajectoires. +\end{rem} + +\begin{defin} +Un point d'équilibre $x^*$ est stable au sens de Lagrange si et seulement si + +\[\forall \delta > 0, \exists \epsilon > 0 \text{ tel que } \forall t \in \R, || s(t_0,x_0)-x^* || \leq \delta \Rightarrow ||s(t,s(t_0,x_0))-x^* || \leq \epsilon\] +\end{defin} + +Ainsi la stabilité suivant Lagrange est qu'un petit changement borné sur $x^*$ implique un petit changement borné après + +\[\forall \delta > 0, \exists \epsilon > 0 \text{ tel que } ||s(t_0,x_0)|| \leq \delta \Rightarrow ||s(t,s(t_0,x_0))|| \leq \epsilon \] + +Sans perte de généralité, on considère le point d'équilibre $x^* = 0$. + +%\img{0.5}{4/lag} + +\paragraph{Stabilité au sens de Lyapunov} + +\begin{defin} +\[\forall \epsilon > 0, \exists \delta > 0 \text{ tel que } ||s(t_0,x_0)|| \leq \delta \Rightarrow || s(t,s(t_0,x_0)) || \leq \epsilon\] +\end{defin} + +Attention : il n'y a pas d'implication entre les deux. + +%\img{0.5}{4/lya} + +\begin{rem} +La condition de Lagrange est sur la bornitude de la trajectoire (quelles que soient les conditions initiales, on borne la solution). Par contre, la condition de Lyapunov est sur la convergence dans un voisinage (il existe des conditions initiales pour lesquelles les trajectoires convergent vers $x^*$). +\end{rem} + +\begin{example}[Oscillateur de Van der Pol] +\[ + \begin{cases} +\dot{x_1} & = x_2\\ \dot{x_2} & = -x_1 + (1-x_ 1^2)x_2 +\end{cases} +\] + +Point d'équilibre $x^* =(0,0)$ + +\begin{rem} +Il n'existe pas de solution analytique aux équations de Van der Pol, mais numériquement on trouve un cycle limite stable. +\end{rem} + +%\img{0.3}{3/2.png} + +$\exists \epsilon$ tel que le cycle limite $\subset$ cercle de centre (0,0) et de rayon $\epsilon$ : stable au sens de Lagrange.\\ + +Par contre, pas stable au sens de Lyapunov car on a +$ \forall \delta > 0, \nexists \epsilon > 0 \text{ tel que } ||s(t,s(t_0,x_0))|| < \epsilon $ + +\end{example} + +\begin{example}[Pendule sans frottement] +L'origine est stable suivant Lyapunov avec $\delta = \epsilon$. + +Elle n'est pas stable suivant Lagrange $x_0=(x_1= \pi, x_2=0)$ : $\nexists \epsilon >0 \text{ tel que } ||s(t,s(0,s_0))|| < \epsilon$ +\end{example} + +\paragraph{Stabilité uniforme} +\begin{defin} +Le point d'équilibre $x^* (x^* =0)$ est dit point d'équilibre uniformément stable si, pour la condition de Lyapunov, $\delta$ peut être choisi indépendamment des conditions initiales $t_0,x_0$ +\end{defin} + + +\begin{defin} +On définit les classes suivantes : +\begin{enumerate} +\item Si $\alpha : \R_+ \rightarrow \R_+$ est continue et strictement croissante, $\alpha$ est dite de classe $\Kc$. + +Si $\alpha$ croit indéfiniment (i.e. $\alpha (s) \rightarrow \infty$), alors $\alpha\in \Kc_{\infty}$ + +\item $\phi$ est dite de classe $\Lc$ si $\phi:\R_+\rightarrow\R_+$ continue, strictement décroissante et $\phi(s) \rightarrow 0$ + +\item $\beta$ est dite de classe $\Kc\Lc$ si $\beta:\R_+ \times \R_+ \rightarrow \R_+$ si $\beta(.,r)\in \Lc \text{ et } \beta(s,.) \in \Kc$ + +Typiquement $\beta(s,r)=\alpha(s).\phi(r) \text{ avec } \alpha\in\Kc, \phi \in \Lc$. +\end{enumerate} +\end{defin} + +\begin{example} +$\beta(||x_0||,|t|)=||x_0||e^{-\lambda |t|} \text{ avec } \lambda >0$ + +Ainsi le but est d'arriver à vérifier pour une trajectoire du système $ ||s(t,x_0)|| \leq \beta(||x_0||,t),t \geq 0$ (enveloppe) +\end{example} + +\begin{prop} +L'origine est uniformément stable si et seulement si \[\exists c>0, \alpha \in \Kc \text{ tel que } ||s(t_0,x_0)|| \leq c \Rightarrow ||s(t,s(t_0,x_0))|| \leq \alpha (||s(t_0,x_0)||)\] +\end{prop} + +\begin{proof} +Condition suffisante. + +Soit $\alpha \in \Kc$ (strictement croissante et continue, donc $\alpha^{-1}$ existe). + +Pour $\epsilon >0, \exists \delta$ dépendant de $\epsilon \text{ tel que } \delta = \alpha^{-1}(\epsilon)$. + +Si $||s(t_0,x_0)|| \leq \delta \Rightarrow ||s(t,s(t_0,x_0))|| \leq \alpha(\alpha^{-1}(\epsilon)) \leq \epsilon$\\ + +Condition nécessaire. + +$\forall \epsilon>0, \exists \delta$ dépendant de $\epsilon \text{ tel que } ||s_0|| \leq \delta \Rightarrow ||s|| \leq \epsilon$ + +Si $\epsilon_2 > \epsilon_1 \Rightarrow \delta_2 \geq \delta_1$ (suivant Lyapunov). On définit $\delta' \in \Kc \text{ tel que } \delta'<\delta$. + +Pour $\epsilon > 9, \exists \delta > 0 \text{ tel que }$ +\begin{align*} +||s_0|| \leq \delta & \Rightarrow ||\delta|| \leq \epsilon\\ +||s_0|| \leq \delta' & \Rightarrow ||\delta|| \leq \epsilon \text{ car } \delta'<\delta +\end{align*} + +Si on définit $\alpha(||.||)=(\delta')^{-1}$, $\forall \epsilon >0, \exists \delta'(\epsilon)$ où $||s_0||=\delta'(\epsilon) \Rightarrow \epsilon = (\delta')^{-1}(||s_0||)$ + +Suivant Lyapunov, cela implique $||s|| \leq \epsilon \leq \alpha (||s_0||)$ +\end{proof} + +\paragraph{Attractivité (convergence)} +\begin{defin} +$\exists r > 0, \forall \sigma > 0, \exists T > 0 \text{ tel que } ||s(t_0,x_0)|| \leq r \Rightarrow ||s(t,s(t_0,x_0))|| \leq \sigma, \forall t \geq T$ + +%\img{0.5}{4/1.png} + +Autrement dit : $||s_0|| \leq r \Rightarrow \lim_{t\rightarrow \infty} ||s_t|| = 0$. + +On parle d'attractivité uniforme si $T$ ne dépend pas de $t_0$. +\end{defin} + +\paragraph{Stabilité asymptotique} +L'origine est asymptotiquement stable si et seulement si +\begin{itemize} +\item stabilité au sens de Lyapunov et attractivité +\item $||s_0|| \leq r \Rightarrow ||s|| \leq \beta (||s_0||,t), \quad \beta \in \Kc\Lc$ +\end{itemize} + +\paragraph{Stabilité exponentielle} +L'origine est exponentiellement stable si et seulement si +\begin{itemize} +\item stabilité au sens de Lyapunov et attractivité +\item $\exists \alpha, \lambda, r >0 \text{ tel que } ||s_0|| \leq r \Rightarrow ||s|| \leq \alpha ||s_0|| e^{-\lambda t}$ +\end{itemize} + +\paragraph{Stabilité locale et globale} +\begin{itemize} +\item L'origine est globalement stable si la stabilité (asymptotique, exponentielle,...) ne dépend pas de la condition initiale, i.e. $\forall t_0 \in \R \text{ et } x_0 \in \R^n$ et dit localement stable (asymptotiquement, exponentiellement,...) +\item Si la stabilité dépend de la CI, i.e. $\exists V_t \subset \R$ ou $V_x \in \R^n$ tel que $\forall t_0 \in V_t$ et $\forall x_0 \in V_x$, l'origine est stable. +\end{itemize} + +\paragraph{Problème} Généralement, on n'a pas de solution analytique de l'équation différentielle. Ainsi, la stabilité ne peut pas être vérifiée via la trajectoire. + +\begin{defin}[Fonction de Lyapunov] +$V$ est une fonction de Lyapunov si : +\begin{enumerate} +\item $V : + \begin{cases} +\R^n & \rightarrow \R_+\\x & \mapsto V(x) +\end{cases} +$ telle que $V(0)=0$ et $V(x) \geq 0$ (définie semi-positive) ou telle que $V(0)=0$ et $V(x) > 0$ si $x\neq 0$ (définie positive) +\item $V$ est radialement non bornée, i.e. $V(x) \rightarrow_{||x|| \rightarrow \infty} \infty$ +\end{enumerate} +\end{defin} + +\begin{thm}[Stabilité au sens de Lyapunov] +Soit $\dot{x}(t) = f(x(t))$ et $f(0)=0$ (origine est un point d'équilibre). On suppose qu'il existe $V$ (fonction de Lyapunov) continue et différentiable tel que +\[ \exists D \subset \R^n, 0 \in \D \text{ où } \forall x \in \D, \quad \dot{V}(x) = (\derivp[V]{x})^Tf(x) \leq 0 \] +Alors l'origine est stable au sens de Lyapunov sur $\D$. + +Si $\D = \R^n$, 0 est globalement stable au sens de Lyapunov. +\end{thm} + +\begin{proof} +Si $x=0$ est stable, alors $\forall \epsilon > 0, \exists \delta > 0 \text{ tel que } ||s_0|| \leq \delta \Rightarrow ||s|| \leq \epsilon$. + +Pour $\epsilon > 0$ on définit $0 0 \text{ tel que } A^TP + PA$ définie négative. + +On pose $P = \int_0^{\infty} e^{A^Tt}Qe^{At} dt$ avec $Q$ définie positive. On a donc $P$ définie positive. + +\[ \int_0^{\infty} (A^T e^{A^Tt} Q e^{At} + e^{A^T t} Q e^{At} A)dt = \int_0^{\infty} \dd{e^{A^Tt} Q e^{At}}{t} dt = [e^{A^Tt}Qe^{At}]_0^{\infty}\] + +Si $A$ est Hurwitz : $e^{At} \rightarrow_{t\rightarrow \infty} 0$ + +\[A^T P + PA = -Q \text{ définie négative (équation de Lyapunov)} \] + +Pour le système linéaire +\[ \dot{V}(x) = x^T (A^T P + PA)x \leq -x^T Q x\] +$\Rightarrow$ Stabilité de Lyapunov $\Leftrightarrow$ Stabilité asymptotique +\end{example} + +\begin{thm}[Stabilité exponentielle] +Soient le système $G: \dot{x}=f(x)$ et $f(0)=0$, $\exists V : \D \rightarrow \R_+$ fonction de Lyapunov continue et différentiable telle que +\begin{enumerate} +\item $\exists \alpha > 0, \beta > 0$ et $c\geq 1$ tel que +\[ \quad \forall x \in \D, \quad \alpha ||x||^c \leq V(x) \leq \beta ||x||^c\] +\item $\exists \gamma > 0$ tel que +\[ \quad \forall x \in \D, \dot{V} \leq - \gamma V \leq - \gamma ||x||^c \] +\end{enumerate} +Alors l'origine est exponentiellement stable. Si $\D=\R^n$, on a aussi la stabilité globale. +\end{thm} + +\begin{proof} +$\dot{V} \leq -\gamma V \Rightarrow V(x(t)) \leq V(x(0))e^{-\gamma t}$ + +si $\dot{\hat{V}}=-\gamma \hat{V}$ +\begin{align*} +V(x(0)) & \leq \beta ||x(0)||^c \\ \text{ et } V(x(t)) & \geq \alpha ||x(t)||^c \\ +V(x(0))e^{-\gamma t} & \geq \\ +\beta||x(0)||^c e^{-\gamma t} & \geq \qquad \Rightarrow ||x(t)|| \leq (\frac{\beta}{\alpha})^{1/c}||x(0)||e^{-\frac{\gamma}{c}t} +\end{align*} +\end{proof} + +\begin{example} + +Soit le système NL + +\[ + \begin{cases} +\dot{x_1} & = -x_1^3 + x_2 ^3 + x_1x_2^2\\\dot{x_2} & = - x_2^2 x_1 - 5x_2^3 +\end{cases} +\] + +$(x_1,x_2)=(0,0),f(0)=0$ est-il asymptotiquement stable ? + +On pose $V(x) = \frac{1}{2}(x_1^2 + x_2^2)$. $V(0) = 0$ et $V(x)>0, \forall x \neq 0$. + +\[ \dot{V}(x) = x_1\dot{x_1} + x_2 \dot{x_2} = -x_1^4 + x_1^2 x_2^2 - 5 x_2^4 \leq -\frac{1}{2}x_1^4 - \frac{9}{2}x_2^4 \leq - Q(x) \text{ tel que } Q(x) \geq 0 \] + +L'origine est globalement asymptotiquement stable.\\ + +Est-il exponentiellement stable ? + +\[ \alpha ||x(t)||^c \leq V(x(t)) \leq \beta ||x(t)||^c \] + +$\beta=1,\alpha=\frac{1}{4}$ + +\[ \dot{V} \leq - \frac{1}{2}x_1^4 - \frac{9}{2}x_2^4 \leq -\frac{9}{2}(x_1^4 + x_2^4) \] + +Pour $\D = \{ ||x|| \leq 1 \}, x_1^2 +x_2^2 \geq x_1^4 + x_2^4$ donc $-(x_1^2 + x_2^2) \leq -(x_1^4+x_2^4)$ : on ne peut pas borner $\dot{V}$ par $V$. + +Avec ce $V(x)$ on ne peut décider de la convergence exponentielle. +\end{example} + +Si on arrive pas a vérifier la stabilité alors le point d'équilibre (ou l'origine) peut-être instable. Dans ce cas, comment vérifier l'instabilité du point d'équilibre (origine)?\\ + +\begin{thm}[Théorème de Lyapunov d'instabilité] +Soit le système G: $x=f(x)$, $f(0)=0$ et $t\geq 0$.\\ +Si $\exists V : \D \subset \mathbb{R}^n \rightarrow \mathbb{R}_+$ continue, différentiable et définie positive ($0 \in \D$), tel que +\[\forall x \in \D^*, \quad \dot{V}(x) = \left( \frac{\partial V}{\partial x}\right)^T f(x) >0 \] + alors l'origine est instable. +\end{thm} + +\begin{proof} +Instable $\Leftrightarrow$ $\exists \epsilon>0$ tel que $\forall \delta >0$, alors $||x_0|| \leq \delta$ et $||x|| \geq \epsilon$\\ + +$\forall \delta > 0$ soit $r \in ]0;\delta[$ tel que:\\ +$B_r(0) = \{ x\in \D$ tel que $ ||x|| \leq r \}$ est compact.\\ +On pose $\alpha = max_{B_r(0)} V(x)$ et $x_0 \in B_r(0)$\\ +$V(x_0) = \alpha$, ainsi $V(x) - V(x_0) >0$ : +\begin{align*} +\Rightarrow & V(x) > \alpha\\ +\Rightarrow & x \notin B_r(0) \\ +\Rightarrow & x \in B_r^c(0)\\ +\Rightarrow & ||x||> r +\end{align*} +Donc $\exists \epsilon >0$ tel que $||x|| \geq \epsilon > r$ +\end{proof} + +\begin{thm}[Théorème de Barbashin-Krasovsky (Stabilité asymptotique)] +Soit $\{0\}$ un point d'équilibre du système $\dot{x} = f(x)$ , où $f:\D \rightarrow \mathbb{R}^n$, localement lipschitzienne. On suppose qu'il existe $V$ continue, différentiable et définie positive telle que \[\dot{V} \leq 0\] +Soit $S = \{x \in \D$ tel que $\dot{V(x)} = 0\}$. + +Si $x=0$ est le seul élément de $S$, alors l'origine est asymptotiquement stable. +\end{thm} + +\begin{example} +Soit le système : +\[ + \begin{cases} +\dot{x_1} &= -x_1^3 + 2 x_2^3\\\dot{x_2} &= -2x_1x_2^2 +\end{cases} +\] +L'origine est un point d'équilibre.\\ +\begin{align*} +V(x) &= \frac{1}{2}x_1^2 + \frac{1}{2}x_2^2 >0\\ +\dot{V(x)} &= x_1\dot{x_1} + x_2\dot{x_2} = -x_1^4 \leq 0 +\end{align*} +On ne peut pas conclure sur la stabilité asymptotique car $Q(x) = \frac{1}{2}x_1^4$ ne dépend pas de $x_2$. \\ + +On utilise le théorème de Barbashin : +\begin{align*} +S = \{x \in \D \text{ tel que }\dot{V(x)} = 0\} \Rightarrow x_1 = 0\\ +\Rightarrow & \dot{x_2} = 0\\ +\Rightarrow & x_2 = 0\\ +\Rightarrow & S = \{0\}\\ +\Rightarrow & \text{Stabilité asymptotique} +\end{align*} +\end{example} + +\begin{thm}[Principe d'invariance de LaSalle] +Soient $ \dot{x} = f(x)$ avec $f: \D \rightarrow \mathbb{R}^n$, $\Omega$ un compact positivement invariant tel que $\Omega \subset \D$, $V:\D\rightarrow\mathbb{R}_+$ continue, différentiable tel que $\dot{V} \leq 0 $ dans $\Omega$, $E= \{x \in \Omega$ tel que $ \dot{V}=0\}$ et M le plus grand ensemble positivement invariant inclus dans E. + +Alors toute solution $x$ tel que $x_0 \in \Omega$ converge vers M quand $t \longrightarrow \infty$. Autrement dit $\overline{M}$ est l'attracteur. +\end{thm} + + +\begin{example}[Barbashin] +Soit le système \[ + \begin{cases} +\dot{x_1} & =x_2\\ \dot{x_2} & = -h(x_1) - g(x_2) +\end{cases} +\] où $h,g:[-a,a] \rightarrow \R$ avec $h(0)=g(0)=0$ + +et $\forall x \neq 0, \quad x.h(x) >0 \text{ et } x.g(x) >0$.\\ + +L'origine est un point d'équilibre.\\ + +Fonction de Lyapunov candidate : +\[ V(x) = \int_0^{x_1} h(s)ds + \frac{1}{2}x_2^2 \] + +$x_1 = 0$ et $x_2=0 \Rightarrow V(x)=0$ + +$x_1 \neq 0$ ou $x_2 \neq 0 \Rightarrow V(x) > 0$ + +donc $V$ est définie positive.\\ + +\begin{align*} +\dot{V}(x) & = h(x_1) \dot{x_1}+ x_2 \dot{x_2}\\ +& = h(x_1)x_1 - x_2h(x_1) - g(x_1)x_2 \\ +& = -g(x_2)x_2 \leq -Q(x) \text{ définie positive, dépend de } x_1 \text{ et } x_2 +\end{align*} + +Barbashin : + +$E = \{ x \in \R^2, \dot{V}(x) = 0 \}$ + +$\dot{V}(x)=0 \Rightarrow x_2 = 0 \Rightarrow \dot{x_1}=0$ + +$\dot{x_2} = 0 + x_2 = 0 \Rightarrow h(x_1)=0 \Rightarrow x_1 = 0$ + +Alors $E=\{0\}$ stabilité asymptotique globale. +\end{example} + +\begin{example}[Invariance de La Salle] +Soit le système $\dot{x} = ax + u$, $a$ inconnu mais borné. + +$u=-kx$ et $\dot{k}= \gamma x^2, \gamma >0$ + +On pose $x_1=x$ et $x_2=k$ +\[ + \begin{cases} +\dot{x_1} & = ax_1 - x_2x_1 \\\dot{x_2}& = \gamma x_1^2 +\end{cases} \] + +La fonction de Lyapunov candidate +\[ V(x) = \frac{1}{2} x_1^2 + \frac{1}{2\gamma} (x_2-b)^2, \quad \text{ avec } b>a \text{ car $a$ est borné} \] +$V(0,b)=0$ et non pas l'origine + +$V(x) \geq 0, \forall x \in \R^d$ +\begin{align*} +\dot{V}(x) & = x_1 \dot{x_1} + \frac{1}{\gamma}(x_2-b)\dot{x_2} \\ +& = ax_1^2 - x_1^2 x_2 + (x_2-b)x_1^2 \\ +& = x_1^2 (a-b) \leq 0 +\end{align*} + +$E = \{ x \in \R^2, \dot{V}=0 \} = \{ x_1 = 0 \}$ : attracteur + +Pour le système de départ, on veut montrer que $x\to0$ ie..e. $x_1 \to 0$ donc (attracteur) $x_1 \to 0$ +\end{example} + +\section{Extension du théorème de Lyapunov aux systèmes non autonomes, i.e. $\dot{x}=f(t,x)$} + +\begin{defin} +Un système $G : \dot{x}(t) = f(t,x)$, $x(t_0=x_0, \forall t\geq t_0$ avec $f(t,0)=0$, $\forall t \geq 0 \Rightarrow x = 0$ est un point d'équilibre. + +L'origine est stable au sens de Lyapunov si et seulement si +\[ \forall \epsilon > 0 \text{ et } t_0 \geq 0, \exists \delta > 0 \text{ tel que } || S(t_0,x_0) || \leq \delta \Rightarrow || S(t,S(t_0,x_0)) || \leq \epsilon, \forall t \geq t_0 \] +\end{defin} + +\begin{thm}[Théorème de Lyapunov] +L'origine du système $G$ est stable au sens de Lyapunov s'il existe une $V:[0,+\infty[ \times \D \rightarrow \R_+$ continue et différentiable telle que : +\begin{itemize} +\item $V(t,0) = 0, \forall t\geq 0$ +\item $V(t,x) > 0, \forall (t,x) \in \R_+ \times \D \setminus \{0\}$ +\item $\dot{V}(t,x) = \derivp[V(t,x)]{t} + (\derivp[V(t,x)]{x})^Tf(t,x) \leq 0$, $\forall (t,x) \in \R_+ \times \D$ +\end{itemize} + +S'il existe $Q(t,x)$ tel que +\begin{itemize} +\item $Q(t,0)=0, \forall t \geq 0$ +\item $Q(t,x) > 0, \forall (t,x) \in \R_+ \times \D \setminus \{0\}$ +\item $\dot{V}(t,x) \leq - Q(t,x), \forall (t,x) \in \R_+ \times \D$ +\end{itemize} + +Alors l'origine est asymptotiquement stable.\\ + +Si $\exists \alpha > 0, \beta > 0, \gamma > 0 \text{ et } c \geq 1 \text{ tel que }$ +\begin{itemize} +\item $\alpha ||x||^c \leq V(t,x) \leq \beta ||x||^c$ +\item $\dot{V}(,x) \leq - \gamma ||x||^c$ +\end{itemize} + +Alors l'origine est exponentiellement stable. +\begin{rem} +Si $\D = \R^n$ : l'origine est globalement stable +\end{rem} + +\end{thm} + +Les démonstrations sont calquées sur celles du cas autonome, avec $x_1 = t \in \R_ +$, $x_2 = x \in \R^n$, $x_2 = x \in \R^n$ donc $\dot{x_1} = 1$ et $\dot{x_2} = f(x_1,x_2)$ + +\begin{example}[Système linéaire non stationnaire] +$\dot{x}(t) = A(t) x(t)$ et $x(0)=x_0, t \geq 0$ + +Soit $V(t,x)=x^TP(t)x$ où $P(t) > 9, \forall t \in \R_ +$ + +$V(t,0) = 0, \forall t \in \R_+$ et $V(t,x) > 0, \forall (t,x) \in \R_+ \times \R^n \setminus \{0\}$ + +\begin{align*} +& \dot{V}(t,x) = x^T(t) \dot{P}(t) x(t) + x^T(t)A^T(t)P(t)x(t) + x^T(t)P(t)A(t)x(t) \leq 0 \\ +& \Leftrightarrow \dot{P}(t) + A^T(t)P(t) + P(t)A(t) \leq 0 \\ +\end{align*} +Inégalité de Lyapunov dynamique + +Stabilité asymptotique : +\[ P(t)+A^T(t)P(t) + P(t)A(t) = - Q(t) \] +Équation de Lyapunov dynamique + +\[ \lambda_{min}(P(t)) ||x||^{1=c} \leq V(t,x) \leq \lambda_{max}(P(t)) ||x||^{1=c} \] + +$\forall t \in \R_+, \exists \gamma > 0$ +\[ \dot{V}(t,x) \leq -\lambda_{min}(Q(t))||x|| \] stabilité exponentielle +\end{example} + +\begin{rem} +Dans le cas non autonome, la fonction de Lyapunov candidate peut ne pas dépendre du temps, mais elle doit dépendre de toutes les variables d'état. +\end{rem} + +\begin{example} +Soit le système non-linéaire +\begin{align*} +\dot{x_1}(t) & = -x_1^3(t) + \sin \omega t x_2(t) \\ +\dot{x_2}(t) & = - \sin \omega t x_1(t) - x_2^3(t) +\end{align*} +avec $x_1(0) = x_{10}, x_2(0) = x_{20}$ et $t\geq 0$ + +L'origine est bien un point d'équilibre. Est-il asymptotiquement stable ? + +\begin{align*} +V(x) & = \frac{1}{2} x_1^2 + \frac{1}{2}x_2^2 \\ +\dot{V}(x) & = x_1 (-x_1^3 + \sin \omega t x_2) + x_2(-\sin \omega t x_1 - x_2^3) \\ +& = -x_1^4 - x_2 ^4 \leq 0 \text{ : stable } \\ +& \leq - \frac{1}{2}(x_1^4 + x_2^4) = -Q(x) \text{ : globalement aymptotiquement stable } +\end{align*} +\end{example} + +\section{Stabilité entrées-états (SEE)/ Input-States Stability (ISS)} + +Soit le système $ G: \dot{x}=f(x,u)$ où $f:\R^n \times \R^m \rightarrow \R^n$ ($m$ désigne le nombre d'entrées) + +Soit l'origine un point d'équilibre : + +\begin{enumerate} +\item S'il est globalement stable, alors on peur analyser la SEE +\item S'il est localement stable, alors la SEE est locale ($\D \subset \R^n$) +\end{enumerate} + +Dans le cas 1, on analyse la stabilité du système en SEE. Dans le cas 2, on analyse localement ($\D$) la stabilité du système en SEE. + +\begin{defin} +Le système est dit SEE si $\forall u(t)$ et $\forall x_0 \in \R^n$ bornées, il existe une solution $x(t,x_0), \forall t \geq 0$ et $\exists \alpha \in \Kc\Lc$ et $\exists\gamma \in \Kc_{\infty}$ tels que : + +\[ ||x(t,x_0)|| \leq \alpha(||x_0||,t) + \gamma(||u||_{\infty})\] + +où $||u||_{\infty} = \sup_{t\geq0}||u(t)|| = \sup_{t\geq0} (u^Tu)^{1/2}$ + +\end{defin} + +\begin{rem} +\[ \lim_{t \to \infty} ||x(t,x_0)|| \leq \gamma (||u||_{\infty}) \] + +$\gamma$ gain asymptotique du système +\end{rem} + + +\begin{example} +Soit le système $\dot{x}= Ax + Bu$ + +A Hurwitz implique que l'origine est stable. + +Le système est-il SEE ? +\[ x(t,x_0) = e^{At}x_0 + \int_0^t e^{A(t-\tau)} Bu(\tau) de \tau \] + +\begin{align*} +||x(t,x_0)|| \leq e^{\lambda_{min}(A)t}||x_0|| + \frac{1}{k} ||B||.||u||_{\infty} = \frac{1}{k} \gamma(||u||_{\infty}) \text{ où } k = -\lambda_{max}(A) +\end{align*} + +$||B|| = \sup_{||v||=1} ||Bv||$ +SEE +\end{example} + + +\end{document} diff --git a/424-Systeme_Non_Lineaires/Cours/chap5.tex b/424-Systeme_Non_Lineaires/Cours/chap5.tex new file mode 100644 index 0000000..0a624e3 --- /dev/null +++ b/424-Systeme_Non_Lineaires/Cours/chap5.tex @@ -0,0 +1,161 @@ +\documentclass[main.tex]{subfiles} +% Relu jusqu'à 3.4 inclus, X 08/02/2015 +% Corrigé jusqu'au 4.3 inclus, A 28/02/2015. +\newcommand{\Lc}{\mathcal{L}} +\newcommand{\D}{\mathcal{D}} +\begin{document} +\section{Introduction (notations maths)} + +\begin{defin}[Champ de vecteur] +C'est une application de $\R^n \rightarrow \R^n$. +\end{defin} + +\begin{defin}[Crochet de Lie] +Soit $f : \R^n \rightarrow \R^n$ et $g : \R^n \rightarrow \R^n$, on définit le crochet de Lie : +\[ [f,g] : + \begin{cases} +\R^n & \rightarrow \R^n \\ x & \mapsto J_g(x)f(x) - J_f(x)g(x) +\end{cases} +\] +où $J_f$ et $J_g$ sont respectivement les matrices jacobiennes de $f$ et $g$. +\end{defin} + +\begin{prop}[Crochet de Lie] +Soient $f, g \text{ et }h$ des champs de vecteurs et $\lambda_1, \lambda_2 \in \K, (\K = \R \text{ ou } \C)$. +Alors +\begin{align*} +[\lambda_1 f + \lambda_2 g, h ] = \lambda_1[f,h] + \lambda_2[g,h] \quad & \text{Bilinéaire} \\ +[f,g] = - [g,f] \quad & \text{Anti-symétrique} \\ +[f,[g,h]] + [h,[f,g]] + [g,[h,f]] = 0 \quad & \text{Identité de Jacobi} \\ +[f,f] = 0 \quad +\end{align*} +\end{prop} + +\begin{defin}[Algèbre de Lie] +$G$ est une algèbre de Lie sur $\K$ si $G$ est un espace vectoriel ayant pour loi interne le crochet de Lie. +\end{defin} + +\begin{rem} +Cette définition se restreint au cas qui nous intéresse ici, ce n'est pas la définition générale. +\end{rem} + +\begin{rem} +$\Lc(E)$ est l'algèbre de Lie ayant pour famille génératrice l'ensemble des champs de vecteurs $E$. +\end{rem} + +\underline{Notation} : Crochet de Lie itéré + +$ad_f^0 (x) = g(x)$ + +$ad_f^1 g(x) = [f,g](x)$ + +$ad_f^k g(x) = [f,ad_f^{k-1}g](x)$ + +\begin{defin}[Dérivée de Lie] +la dérivée de Lie d'une fonction $\alpha : \R^n \rightarrow \R$ dans la direction de $f : \R^n \rightarrow \R^n$, notée $L_f\alpha$, est définie par : +\[L_f \alpha(x) = \sum_{i=1}^n \derivp[\alpha(x)]{x_i}f_i(x) \] + +Ainsi, +\[L_f^k \alpha (x) = J_{L_f^{k-1} \alpha} (x) f(x) = [ \derivp[L_f^{k-1} \alpha(x)]{x_1} \dots\derivp[L_f^{k-1} \alpha(x)]{x_n}] \vect{f_1(x) \\ \vdots \\ f_n(x) } \] +\end{defin} + +\begin{rem} +\begin{itemize} +\item $L_f^0 (x) = \alpha(x)$ +\item Soient 2 champs de vecteurs $f$ et $g$, alors +\begin{align*} +L_g L_f \alpha (x) & = J_{L_f \alpha}(x) g(x) \\ +L_{[f,g]} \alpha(x) & = L_f L_g \alpha(x) - L_gL_f \alpha(x) +\end{align*} +\end{itemize} +\end{rem} + +\begin{defin}[dimension] +La dimension d'un ensemble de champs de vecteurs $E=\{f_1(x) \dots f_n(x)\}$, où $f_i(x) : \R^n \rightarrow \R^n$, est la dimension de l'espace vectoriel $\Delta(x)$ engendré par l'ensemble $E$. + +\begin{rem} +On fait la confusion entre rang et dimension. +\end{rem} +\end{defin} + +\begin{example} +\[ f_1(x) = \vect{x_1 \\ x_2 \\ 2}, f_2(x) = + \begin{bmatrix} +x_1 & x_3 \\ x_2 & x_3 \\2 & x_3 +\end{bmatrix} +\text{ et }f_3(x) = \vect{x_2 \\ x_2 \\ 0} \] + +Si $x_2 = 0$, alors $\Delta(x) = vect\{( \vect{x_1 \\ 0 \\ 2} ) \} \text{ et }dim=1$. + +Si $x_2 \neq 0$, alors $\Delta(x) = vect\{(\vect{x_1 \\ x_2 \\ 2},\vect{1 \\ 1 \\ 0})\} \text{ et }dim=2$. +\end{example} + +\section{Commandabilité (atteignabilité, contrôlabilité)} + +Soit le système non-linéaire (1) (affine en la commande) +\[ \dot{x} = f(x) + g(x)u = f(x) + \sum_{i=1}^m g_i(x) u_i, \quad x \in \R^n \text{ et }u \in \R^m \] + +\begin{defin}[Commandabilité] +Un système est commandable ssi $\forall x \in \R^n, \exists u$ tel que $x$ est atteignable dans un temps fini. +\end{defin} + +\begin{thm}[Théorème de Commandabilité] +Le système (1) est commandable ssi la sous-algèbre de Lie $\D = \{g_1 \dots g_m, \Lc(E)\}$ avec $E=\{g_1 \dots g_m,f\}$ est de dimension $n$. +\end{thm} + +\begin{example}[linéaire] +\[ \dot{x} = Ax + Bu \] + +\[ E = \{Ax,B\}, [B,Ax] = AB \] +\[ [AB,Ax] = A^2B, \dots, A^{n-1}B, \dots \] +\[ \Lc(E) = vect \{AB,A^2B,\dots\} \] + +suivant Cayley Hamilton: +\[ \D = \{B,vect \{AB,AB^2,\dots,A^{n-1}B\}\}\] + +$dim \D = rang (B AB \dots A^{n-1}B)$ théorème de Kalman +\end{example} + +\section{Observabilité (distingabilité)} +Soit le système NL (2) (affine en la commande) : +\begin{align*} +\dot{x} & = f(x) + g(x)u \\ +y & = h(x) +\end{align*} + +\begin{defin}[Observabilité] +Un système est observable si $\forall x_1,x_2 \in \R^n$ 2 conditions initiales telles que $x_1 \neq x_2$, $\exists$ une commande $u$ admissible telle que les sorties soient distinctes, $\forall t \geq t_0$ ($t_0$ instant initial). +\end{defin} + +\begin{defin}[Espace d'observabilité] +$\mathcal{V}$ est l'espace d'observabilité constitué de toutes les combinaisons linéaires obtenues à partir des dérivées de Lie $L_f$ et $L_g$ des fonctions $h_j(x),j=1 \dots p$ telles que $y\in\R^p$ +\[ \mathcal{V} = \{h_j,L_fh_j, L_g h_j, L^2_f h_j,\dots L_g L_f h_j, L_f L_g h_j,\dots \}\] + +Soit $\nabla \mathcal{V}$ l'ensemble des différentielles (gradient) des éléments de $\mathcal{V}$ : + +\[ \nabla \mathcal{V} = \{ \nabla h_j, \nabla L_f h_j ... \} \] +\end{defin} + +\begin{thm}[Théorème d'observabilité] +Le système (2) est localement observable en $x_0$ si $dim \nabla \mathcal{V}(x_0) = n$ et il est observable si $\forall x \in \R^n, dim \nabla \mathcal{V}(c) = n$ +\end{thm} + +\begin{example}[linéaire] +\begin{align*} +\dot{x} & = Ax + Bu = f(x) + g(x)u\\ +y & = Cx = h(x) +\end{align*} + +\begin{align*} +\mathcal{V} & = \{ h(x), L_fh(x), L_gh, L^2_fh ,L_g^2h , L_fL_gh , L_gL_fh \dots \} \\ +\mathcal{V} & = \{ Cx, C.Ax (=L_fh(x)), C.B (=L_gh), CA^2x (=L^2_fh) ,0 (=L_g^2h) , 0 (=L_fL_gh) , CAB (=L_gL_fh) \dots \} \\ +\nabla \mathcal{V} & = \{ C , CA , 0 CA^2 , 0 , 0 , 0 \dots \} \\ +dim \nabla \mathcal{V} & = rang \vect{ C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1}} \quad \text{Critère de Kalman} +\end{align*} +\end{example} + +\begin{rem} +l'action de la commande intervient dans l'observabilité. Cette contrainte est écartée dnas le cas linéaire. +\end{rem} + +\end{document} diff --git a/424-Systeme_Non_Lineaires/Cours/chap6.tex b/424-Systeme_Non_Lineaires/Cours/chap6.tex new file mode 100644 index 0000000..b9361c3 --- /dev/null +++ b/424-Systeme_Non_Lineaires/Cours/chap6.tex @@ -0,0 +1,783 @@ +\documentclass{article} +\input{../../preambule/preambule} + +\newcommand{\nom}{Commande des systèmes non linéaires} +\newcommand{\partie}{} +\renewcommand{\nomentete}{UE424 - \nom} +\renewcommand{\auteur}{\partie} + + + +\begin{document} + +\titre{\nom} + +\renewcommand{\partie}{Synthèse de commande en non-linéaire} + + +\noindent \textsc{Avertissement : } La structure de ce cours est purement fictionnelle. Toute ressemblance avec un plan existant ou ayant existé serait tout à fait fortuite. + + +\section*{Commande par bouclage linéarisant} + +Principe : se ramener à un comportement linéaire +\img{0.5}{5/1} + +\subsection*{Linéarisation entrées-sorties} + +Cas SISO : $u\in \R$ et $y\in\R$ + +Soit le système NL (1) (affine en la commande) : +\[ \acc{\dot{x} & = f(x) + g(x) u}{y & = h(x)} \] + +\begin{dfn}[degré relatif] +Le degré relatif $r$ du système (1) est défini par : +\[ r \in \N \tq L_gL_f^{r-1}h(x) \neq 0 \et \forall k < r-1, L_gL_f^{k}h(x) = 0\] +\end{dfn} + +\begin{thrm} +[Procédure de linéarisation] + +\begin{align*} +\intertext{On cherche $r$ par le calcul des dérivées successives de $y=h(x)$ :} +\dot{y} & = \drond{h(x)}{x} \dot{x} + \\ +& = \drond{h(x)}{x}(f(x)+g(x)u) \\ +& = L_fh(x) + L_gh(x)u +\intertext{Si $L_gh(x)\neq0$, alors $r=1$. Sinon on continue la procédure :} +y^{(2)} & = \drond{L_fh(x)}{x}\dot{x} \\ +& = \drond{L_fh(x)}{x}f(x) + \drond{L_fh(x)}{x}g(x)u \\ +& = L^2_fh(x) + L_gL_fh(x)u +\intertext{Si $L_gL_fh(x) \neq 0$, alors $r=2$. Sinon on continue...} +y^{(r)} & = L_f^rh(x) + L_gL_f^{r-1}h(x)u +\intertext{On a $1 \leq r \leq n$ car la procédure utilise la base canonique ($x_1=y,x_2=\dot{y}$) : la commande doit apparaître au maximum à la $n$-ième dérivée.} +\intertext{On pose $v=y^{(r)} = L_f^rh(x) + L_gL_f^{r-1}h(x)u$} +u & = (L_gL_f^{r-1}h(x))^{-1}(v-L_f^rh(x)) \\ +& = \alpha(x) + \beta(x) v, \avec \acc{\alpha(x) & = -(L_gL_f^{r-1}h(x))^{-1}L_f^rh(x)}{ \beta(x) & = (L_gL_f^{r-1}h(x))^{-1}} +\end{align*} + +La nouvelle entrée de commande est $v$ telle que +\[ \acc{ \dot{x} & = f(x) + g(x)\alpha(x) + g(x)\beta(x)v}{ y & = h(x)} \] + +$u = \alpha(x) + \beta(x)v$ est le bouclage linéarisant statique car à un instant fixé, la linéarisation ne dépend que de $x$ à cet instant.\\ + +\underline{Cas $r=n$} + +\begin{multicols}{2} +Choix de la base : +\begin{align*} +z_1 & = y = h(x) \\ +z_2 & = \dot{y} = L_fh(x) \Rightarrow \dot{z_1} = z_2 \\ +z_3 & = \ddot{y} = L_g^2h(x) \Rightarrow \dot{z_2} = z_3 \\ +\vdots \\ +y^{(n)} & = \dot{z_n} = L_f^nh(x) + L_gL_f^{n-1}h(x)u = v +\end{align*} + +Nouveau modèle : +\begin{align*} +\dot{z_1} & = z_2 \\ +\vdots \\ +\dot{z_{n-1}} & = z_n \\ +\dot{z_n} & = a(z) + b(z)u = v +\end{align*} +donc \[ u = \frac{v-a(z)}{b(z)} \avec b(z) \neq 0 \] +\end{multicols} + +\[ z = \phi(x) = \vect{\phi_1(x) \\ \vdots \\ \phi_n(x)} = \vect{ h(x) \\ L_fh(x) \\ \vdots \\ L_f^{n-1}h(x)} \] +\[ u = \alpha(x)+\beta(x)v \avec \alpha(x) = -\frac{a(z)}{b(z)}|_{z=\phi(x)} \et \beta(x) = \frac{1}{b(z)}|_{z=\phi(x)} \] +\end{thrm} + +\newpage +Schéma blocs : +\img{0.5}{5/2} + +Modèle linéaire : +\[ +\acc{\dot{z} & = Az + Bv}{y & = Cz} \avec A = \left[ \begin{array}{ccccc} +0 & 1 & 0 & \dots & 0 \\ +\vdots & \ddots & \ddots & \ddots & \vdots \\ +\vdots & & \ddots & \ddots & 0 \\ +0 & \dots & \dots & 0 & 1 \\ +0 & \dots & \dots & \dots & 0 +\end{array} \right], \quad B = \vect{ 0 \\ \vdots \\ 0 \\ 1 } \et C = [1 \quad 0 \dots 0 ] +\] + +Synthèse du correcteur linéaire : +\img{0.5}{5/3} + +Planification de trajectoire : +\[ y^{(n)} = v = y_c^{(n)} + a_1 (y_v^{n-1)} - y^{(n-1)})+ \dots + a_{n-1}(\dot{y_c} - \dot{y}) + a_n(y_c-y) \] + +Les $a_i$ sont choisis en imposant la dynamique de $\epsilon=y-y_c$ : +\[ \epsilon^{(n)} + a_1 \epsilon^{(n-1)} + \dots + a_{n-1}\dot{\epsilon} + a_n\epsilon = 0 \] + +Matrice d'évolution de la boucle fermée : +\[ A_{BF} = \left[ \begin{array}{ccccc} +0 & 1 & 0 & \dots & 0 \\ +\vdots & \ddots & \ddots & \ddots & \vdots \\ +\vdots & & \ddots & \ddots & 0 \\ +0 & \dots & \dots & 0 & 1 \\ +-a_n & -a_{n-1} & \dots & \dots & -a_1 +\end{array} \right] \quad \text{Forme canonique} +\] + +\begin{rmq} +Cette méthode est assez simple. Cependant, il faut accéder aux dérivées successives de la sortie. Si on a des capteurs, alors OK, mais calculer une dérivée numérique n'est pas génial. +\end{rmq} + +\subsection*{Linéarisation entrée-états} + +On ne dispose pas d'une sortie $y=h(x)$ donc on essaye de trouver une sortie "fictive".\\ + +Problème : trouver le bon changement de base $z_1 = \phi_1(x)$ qui remplace $z_1=y=h(x)$ : +\[ z = \vect{z_1 \\ \vdots \\ z_n} = \vect{\phi_1(x) \\ \vdots \\ \phi_n(x)} = \phi(x) \] + +$\phi$ est un difféomorphisme, i.e. bijectif et différentiable, de même pour la réciproque.\\ + +\cacededi{Donc on va définir c'est quoi la linéarisation.}{M. Abbas Turki} + +\begin{dfn} +Le système $\dot{x} = f(x) + g(x)u$ (1) est linéarisable entrée-états si il existe une région $\Omega \in \R^n$, un difféomorphisme $\phi:\Omega\rightarrow\R^n$ et un retour d'état $u=\alpha(x) + \beta(x)v$ tels que le nouveau vecteur d'état est $z=\phi(x)$ et la nouvelle entrée est $v$ avec $\dot{z} = Az+Bv$, $A$ matrice d'évolution $\in \R^{m \times n}$. +\end{dfn} + +En s'inspirant de la linéarisation entrée-sortie, on simplifie la recherche de $\phi(x)$ par celle de $\phi_1(x)=z_1$ et le reste des transformations est obtenu par la forme canonique (forme normale). + +\begin{align*} +z_2 & = \phi_2(x) \\ +& = \dot{z_1} = \dot{\phi_1}(x) \\ +& = \drond{\phi_1(x)}{x}f(x) + \drond{\phi_1(x)}{x}g(x)u \\ +& = L_f\phi_1(x) + L_g\phi_1(x)u \avec L_g\phi_1(x) = 0 \\ +z_3 & = L_f^2 \phi_1(x) \avec L_gL_f\phi_1(x) = 0 \\ +\phi(x) & = \vect{\phi_1(x) \\ L_f\phi_1(x) \\ \vdots \\ L_f^{n-1} \phi_1(x)} \avec L_gL_f^j \phi_1(x) = 0, j = 0,\dots n-2 +\end{align*} + +Or, $ L_gL_f^j \phi_1(x) = 0, j = 0,\dots n-2 \Leftrightarrow L_{ad_f^j g} \phi_1(x) = 0 $ car +\begin{align*} +L_g(L_g\phi_1) - L_g(L_f\phi_1)) & = L_f(\drond{\phi_1}{x}g)-L_g(\drond{\phi_1}{x}f) \\ +& = \drond{^2\phi_1}{x^2}g.f + \drond{\phi_1}{x}J_g.f - \drond{^2 \phi_1}{x^2}g.f - \drond{\phi_1}{x}J_f.g \\ +0 & = L_{[f,g]}\phi_1 = L_{ad_f g} \phi_1 = 0 +\end{align*} + +Existe-t-il $\phi_1(x)$ tel que +$\acc{L_g L_f^j \phi_1(x) & = 0, \quad j = 0, \dots, n-2}{L_g L_f^{n-1} \phi_1(x) & \neq 0}$ ? + +\begin{dfn}[Distribution de champs de vecteurs] +L'application $\Delta(x)$ est une distribution de champs de vecteurs sur $\Omega$ si $\forall x \in \Omega, \Delta(x)$ est un sous-espace vectoriel. +\end{dfn} + +\begin{exemple} +\[\Delta(x) = vect \left\lbrace \vect{x_1 \\ x_2 \\2}, \vect{x_1x_3 \\ x_2x_3 \\ 2x_3},\vect{x_2 \\ x_2 \\ 0} \right\rbrace \] + +\begin{align*} +x_2 = 0 & \Rightarrow \Delta(x) = vect \left\lbrace \vect{x_2 \\ 0 \\ 2} \right\rbrace \\ +& \Rightarrow \Delta(x) \text{ est e.v. de dim = 1} \\ +x_2 \neq 0 +& \Rightarrow \Delta(x) = vect \left\lbrace \vect{x_1 \\ x_2 \\ 2},\vect{1 \\ 1 \\ 0} \right\rbrace \\ +& \Rightarrow \Delta(x) \text{ est un e.v. de dim = 2} +\end{align*} +$\Delta(x)$ est une distribution de champs de vecteurs. +\end{exemple} + +\cacededi{On a défini c'est quoi une distribution de champs de vecteurs, on va définir c'est quoi une distribution involutive}{M. Abbas-Turki} + +\begin{dfn}[Involution] +La distribution $\Delta$ est involutive ssi +\[\forall f,g \in \Delta, \quad [f,g] \in \Delta, \quad \text{Homogénéité est mère de vertu} \] +\end{dfn} + +\begin{rmq} +$\Delta(x) = vect \{ f_1, \dots, f_p \}$ est une distribution involutive ssi +\[ \exists \alpha_{ij_k} : \R^n \mapsto \R \tq [f_i,f_j] = \sum_{k=1}^p \alpha_{ij_k}(x) f_k, \quad i = 1,\dots p, j = 1,\dots p\] +\end{rmq} + +\begin{exemple} +\[ x_2 \neq 0 \Rightarrow \Delta(x) = vect\left\lbrace \vect{x_1 \\ x_2 \\ 2}, \vect{1 \\ 1 \\ 0} \right\rbrace\] +\begin{align*} +[f_1,f_2] & = J_{f_2}f_1 - J_{f_1} f_2 \\ +& = \mat{ 0 & 0 & 0 }{ 0 & 0 & 0 }{ 0 & 0 & 0 } \vect{x_1 \\ x_2 \\ 2} - \mat{ 1 & 0 & 0 }{ 0 & 1 & 0 }{ 0 & 0 & 0 }\vect{ 1 \\ 1 \\ 0} \\ +& = \vect{ -1 \\ -1 \\ 0} = -f_2 \in \Delta +\end{align*} + +$\Delta$ est une distribution involutive pour $x_2 \neq 0$ +\end{exemple} + +\newcommand{\lesys}{$\dot{x}=f(x)+g(x)u, x\in\R^n, u\in\R$} + +\begin{thrm}[Théorème d'existence de $\phi_1$] +Soit le système \lesys. + +Il existe un changement de base $z=\phi(x)$ linéarisant sur $\Omega$ tel que $\phi^T(x) = [\phi_1(x), L_f\phi_1(x) \dots L_f^{n-1} \phi_1(x) ]$ ssi : +\begin{itemize} +\item $dim(g,ad_f g, \dots ad_f^{n-1} g) = n$ (Commandabilité Kalman) +\item la distribution engendrée par $\{g, ad_f g, \dots ad_f^{n-1}g\}$ est involutive, $\forall x \in \Omega$ \footnote{Homogénéité est mère de vertu} +\end{itemize} +\end{thrm} + +Ainsi, la procédure de linéarisation entrée-états est réalisée via les étapes suivantes : +\begin{enumerate} +\item Construction de $E = \{ g, ad_f g, \dots, ad_f^{n-1} g \}$ +\item Vérifier la commandabilité, i.e. $dim(E) = n$ (Kalman !) +\item Montrer que $\Delta(x) = vect\{E\}$ est involutif, i.e. $\exists \alpha_{ij_k}(x) : \Omega \mapsto \R$ tel que : +\[ [ad_f^i g, ad_f^j g] = \sum_{k=0}^{n-1} \alpha_{ij_k}(x).ad_f^k g, \quad i,j = 0,\dots,n-1 \] +\item Trouver $\phi_1(x)$ avec $\acc{L_{ad_f^j g} \phi_1(x) & = 0, j=0,\dots n-2}{L_gL_f^{n-1} \phi_1(x) & \neq 0}$ +\item Construction du nouveau vecteur d'état $z^T = \phi^T(x) = [\phi_1(x), L_f\phi_1(x) \dots L_f^{n-1} \phi_1(x) ]$ +\item Linéarisation par retour d'état statique $u = \alpha(x)+\beta(x)v$, $v$ nouvelle commande du modèle linéaire, avec +\[ \alpha(x) = -\frac{L_f^n \phi_1(x)}{L_gL_f^{n-1}\phi_1(x)} \et \beta(x) = \frac{1}{L_gL_f^{n-1}\phi_1(x)} \] +\end{enumerate} + +Si le degré relatif $r>1, \quad H_0(p) = \frac{H_1(p)}{1+KH_1(p)} \approx \frac{1}{K} \] +\end{exemple} + +\newpage +\section*{Commandes hiérarchisées} + +\subsection*{Échelles de temps} + +Soit le système (1) $\acc{\dot{x_1} & = \epsilon f_1(x_1,x_2,u)}{\dot{x_2} & = f_2(x_1,x_2,u)}$ avec $f_1$ et $f_2$ lisses (de classe $C^{\infty}$), avec $x_1\in\R^{n_1}$ et $x_2\in\R^{n_2}$. + +On suppose que $0 < \epsilon << 1$. + +On suppose $\tau = \epsilon t$ nouveau temps: $\tau$ est plus lent que $t$. + +Ainsi, le système (1) dans la nouvelle échelle temporelle est donnée par + +\begin{align*} +\dd{x_1}{\tau} & = f_1(x_1,x_2,u) \text{ Dynamique lente est d'ordre 0 en } 1/\epsilon \\ +\dd{x_2}{\tau} & = \frac{1}{\epsilon} f_2(x_1,x_2,u) \text{ Dynamique rapide est d'ordre 1 en }1/\epsilon +\end{align*} + +Ainsi, dans le cas d'un point d'équilibre stable, $x_2$ converge rapidement vers le voisinage dépendant de $\epsilon$, d'un point d'équilibre de $f_2=0$. + +\subsection*{Détermination du voisinage} +Pour $\epsilon = 0$, les points d'équilibre du système (1) forment la variété : +\[ \Sigma_0 = \{ (x_1,x_2,u)/ f_2(x_1,x_2,u) = 0 \} \] + +alors que pour $\epsilon \neq 0$, les points d'équilibre forment la variété +\[ \Sigma_{\epsilon} = \{ (x_1,x_2,y) / f_1(x_1,x_2,u) = 0 \et f_2(x_1,x_2,y) \neq 0 \} \] + +On a $\Sigma_{\epsilon} \subset \Sigma_0$ donc $\Sigma_{\epsilon}$ dégénère en $\Sigma_0$ pour $\epsilon=0$. + +\img{0.5}{7/2} + +L'objectif est d'avoir seulement à faire converger $x_1 \tdv x_1^*$.\\ + +À partir du théorème des fonctions implicites, nous avons l'existence de $X_2 \tq x_2 = X_2(x_1,u)$. + +\begin{dfn} +On définit la variété +\[ \Sigma_{0,\epsilon} = \{ (x_1,x_2) / f_2(x_1,x_2,u,\epsilon) = 0 \} \] +avec $\dot{u} = \epsilon v$ où $v$ est une fonction bornée. +\end{dfn} + +La variété $\Sigma_{0,\epsilon}$ est obtenue à partir de $\Sigma_0$ avec une faible variation de la commande. + +\begin{prop} +Soit le système (1) avec $rang(\drond{f_2}{x_2}) = n_2$, alors $\exists X_2(x_1,u,\epsilon)$ tel que $\forall u$ vérifiant $\dot{u} = \epsilon v$, $v$ bornée, $(x_1,x_2 \in \Sigma_{0,\epsilon}$ avec $x_2 = X_2(x_1,u,\epsilon)$. +\end{prop} + +Interprétation : +La variété $\Sigma_0 \Leftrightarrow x_2=X_2(x_1,u)$, obtenue pour $\epsilon=0$, continue d'exister pour $\epsilon \neq 0$ et suffisamment petit si $\dot{u}=\epsilon v$, $v$ bornée. + +\begin{exemple} +[MMC] +\[ \acc{ +L\dd{i}{t} & = u-Ri - k\omega \text{ Dynamique électrique}}{ +J\dd{\omega}{t} & = Ki - \alpha\omega -C_r \text{ Dynamique mécanique temps lent}}\] + +On pose $\epsilon = L << 1$, donc le temps rapide $\tau = \frac{t}{\epsilon}$\\ + +Identification avec le système (1) : $x_1=\omega$ et $x_2 =i$. + +Pour $\epsilon=0$, $\Sigma_0$ est donnée +\[ i = \frac{u-k\omega}{R} = X_2(x_1,u) \] + +Ainsi, la dynamique lente est donnée par +\[ J \dd{\omega}{\tau} = \epsilon(k(\frac{u-k\omega}{R}) - \alpha \omega - C_r) \] + +En temps lent, la nouvelle expression est +\[ J \dd{\omega}{t} = -(\frac{k^2}{R}+\alpha) \omega - C_r + \frac{k}{R}u \] + +On peut améliorer l'approximation de la variété $\Sigma_{0,\epsilon}$ via un DL du 1er ordre. + +\[ i = \frac{u-k\omega}{R} + \frac{L}{R}(\dot{u}-\frac{k}{J}(k(\frac{u-k\omega}{R})-\alpha\omega-C_r)) + \mathcal{O}(L^2) \] + +Par exemple, si on veut avoir $i_0=0$, alors $\Sigma_0 = k\omega$. Pour garder $i_0=0$ pour $\Sigma_{0,\epsilon}$, on doit imposer une variation lente de $u$ (lente par rapport à $L\dd{}{t}$ +\[ \dot{u} = -\frac{k}{J}(\alpha\omega+C_r) - \mathcal{O}(L^2) \] +\end{exemple} + +\begin{rmq} +$C_r = -\frac{\dot{u}J}{k}$ est utilisée pour estimer $C_r$ en modulant $\dot{u}$ afin que $i_0$ reste aussi plat que possible et $\omega=0$. +\end{rmq} + +\subsection*{Hiérarchisation par commande à grand gain} + +Soit le système (1), où la commande n'intervient que sur $x_2$ linéairement : +\[ +\acc{ \dot{x_1} & = f_3(x_1,x_2) }{ \dot{x_2} & = f_2(x_1,x_2) + u }, \quad x_1 \in \R^{n_1}, x_2 \in \R^{n_2}, u \in\R^{n_2} +\] + +Soit $x_2^*$ la trajectoire consigne à imposer à $x_2$. Avec comme hypothèse $f_2(x_1,x_2)$ bornée, nous appliquons la commande à grand gain +\[ u = -\frac{k}{\epsilon}(x_2-x_2^*)\] +où $\epsilon<< 1$ et $k$ matrice diagonale définie positive. + +Ainsi, suivant la nouvelle échelle de temps $\tau = \frac{t}{\epsilon}$ +\[ \acc{ +\dd{x_1}{\tau} & = \epsilon f_1(x_1,x_2) \quad \text{dynaique lente}}{ +\dd{x_2}{\tau} & = \epsilon f_2(x_1,x_2) - k(x_2-x_2^*) \quad \text{perturbation et dynamique de convergence rapide} +}\] + +$\Sigma_0$ est la variété $x_2 = x_2^*$. + +Pour $\epsilon\neq 0$, $\Sigma_{0,\epsilon}$ est la variété $x_2=x_2^*+k\epsilon f_2(x_2,x_2^*)$ + +La dynamique lente est $\dd{x_1}{\tau} = \epsilon f_1(x_1,x_2^*)$. Par conséquent la consigne $x_2^*$ (commande fictive) peut servir à commander la dynamique lente. + +\newpage +\section*{Commande par backstepping} + +Soit un système sous forme triangulaire (apparition successive des différentes commandes) : +\begin{align*} +\dot{x_1} & = f_1(x_1) + x_2 \\ +\dot{x_2} & = f_2(x_1,x_2) + x_3 \\ +& \vdots \\ +\dot{x_n} & = f_n(x_1,\dots x_n) + u +\end{align*} + +\subsection*{Procédure de synthèse} + +\paragraph{Étape 1} Afin d'imposer la consigne $x_1^*$, on utilise la fonction de Lyapunov +\[ V_1(x_1) = \frac{1}{2}(x_1 - x_1^*)^2 \] +Pour assurer la stabilité, il faut que $\dot{V_1}(x_1)$ soit définie négative. +\begin{align*} +\dot{V_1}(x_1) & = (x_1-x_1^*)(\dot{x_1} - \dot{x_1^*}) \\ +& = (x_1 - x_1^*)(f_1(x_1) + x_2 - \dot{x_1^*}) +\intertext{On cherche donc $x_2$ pour que} +\dot{V_1}(x_1) & = \alpha_1(x_1-x_1^*)^2 \quad \avec \alpha_1 < 0 \\ +x_2^* & = \alpha_1(x_1-x_2^*) - f_1(x_1) + \dot{x_1^*} +\end{align*} + +Cela assure la convergence asymptotique de $x_1$ vers $x_1^*$. + +\paragraph{Étape 2} Faire converger $x_2$ vers $x_2^*$. On utilise la nouvelle fonction de Lyapunov +\[ V_2(x_1,x_2) = \frac{1}{2}(x_1-x_1^*)^2 + \frac{1}{2}(x_2-x_2^*)^2 \] +On veut $\dot{V_2}(x_1,x_2)$ définie négative : +\begin{align*} +\dot{V_2}(x_1,x_2) & = (x_1-x_1^*)(\dot{x_1} - \dot{x_1^*}) + (x_2-x_2^*)(\dot{x_2} - \dot{x_2^*}) \\ +& = \alpha_1(x_1-x_1^*)^2 + \alpha_2(x_2-x_2^*)^2 +\intertext{Pour avoir une hiérarchisation dynamique, on pose $\alpha_2 < \alpha_1 < 0$ (la dynamique 2 est plus rapide que la 1)} +(x_2-x_2^*)(\dot{x_2} - \dot{x_2^*}) & = \alpha_2(x_2-x_2^*)^2 \\ +(x_2-x_2^*)(f_2(x_1,x_2) + x_3 - \dot{x_2^*}) & = \alpha_2(x_2-x_2^*)^2 \\ +x_3^* & = \alpha_2(x_2-x_2^*) - f_2(x_1,x_2) + \dot{x_2^*} +\end{align*} +La démarche est la même à l'étape $n$ : +\[ u = \dot{x_n^*} - f_n(x_1,\dots,x_n) + \alpha_n(x_n-x_n^*) \] +avec $\alpha_n < \alpha_{n-1} < \dots < \alpha_2 < \alpha_1$ + +\begin{rmq} +Cette méthode est généralisable à des systèmes sans forme : +\begin{align*} +\dot{x_1} & = f_1(x_1) + g_1(x_1)x_2 \\ +\dot{x_2} & = f_2(x_1,x_2) + g_2(x_1,x_2)x_3 \\ +& \vdots \\ +\dot{x_n} & = f_n(x_1,\dots,x_n) + g_n(x_1,\dots,x_n)u +\end{align*} +sur $\D = \{x_1,\dots,x_n \tq g_1 \neq 0,\dots,g_n\neq 0 \}$ +\end{rmq} + + +\newpage +\section*{Rejet de perturbation} + +\subsection*{Cas SISO} +\[ (1) \acc{ \dot{x} & = f(x) + g(x)u + p(x) w }{ y & = h(x) } \] + +Même principe que pour la linéarisation par bouclage, on dérive la sortie par rapport au temps : +\[ \dot{y} = \drond{h(x)}{x} \dot{x} = L_fh(x) + L_gh(x) u + L_ph(x) w \] + +\paragraph{Cas 1} $L_ph(x) \neq 0$\\ +Si $L_gh(x) \neq 0$ et la perturbation $w$ est mesurable (rarement), alors le rejet de la perturbation est obtenu par +\[ u = (L_gh(x))^{-1}(v-L_fh(x) - L_ph(x)w) \quad \text{avec trivialement } v = \dot{y}\] +Si la perturbation n'est pas mesurable, on réalise une linéarisation dynamique avec $x_{n+1} = u$ et $x_{n+2} = w$ mais dans ce cas la perturbation $w$ doit être canonique, i.e. $\exists \alpha \in \N \tq w^{(\alpha)} = 0$. + +Ainsi, on dérive la sortie jusqu'à disparition de la perturbation puis on linéarise.\\ + +\noindent Si $L_gh(x) = 0$, on calcule les dérivées d'ordres supérieurs de la sortie jusqu'à apparition de la commande (linéarisation dynamique). + +\paragraph{Cas 2} $L_ph(x) = 0$.\\ +Si $L_gh(x)\neq0$, la perturbation est rejetée pour +\[ u = (L_gh(x))^{-1}(v-L_fh(x)) \] +Su $L_gh(x) = 0$, on dérive une deuxième fois la sortie. + +\begin{prop} +Soient $r$ le degré relatif correspondant à $L_gL_f^{r-1}h(x) \neq 0$ et $\sigma$ le plus petit entier pour lequel $L_pL_f^{\sigma-1}h(x) \neq0$, alors : +\begin{itemize} +\item si $r<\sigma$ la perturbation $w$ est rejetée par la commande linéarisante +\item si $r=\sigma$ la perturbation $w$ est rejetée si elle est mesurable +\item si $r>\sigma$ le rejet de $w$ ne peut se faire que par une linéarisation dynamique : observateur NL si $w$ n'est pas canonique +\end{itemize} +\end{prop} + + +\subsection*{Cas MIMO} +\[ \acc{ \dot{x} & = f(x) + \sum_{i=1}^m g_i(x)u_i + p(x) w }{ y & = h(x) }, \quad x \in \R^n, u \in \R^m, y \in \R^d \] +Même principe que le cas SISO mais une linérisation MIMO où chaque nouvelle entrée $v_i$, permet de rejeter les perturbations sur $y_i$. + +\begin{rmq} +L'incertitude sur le modèle peut être interprétée comme une perturbation. En effet, le modèle (1) s'écrit +\[ \acc{ f(x) & = f(x) + \Delta f(x) + g(x) u + \Delta g(x) u }{ y & = h(x) } \] + +Suivant l'analyse sur le bouclage linéarisant, le rejet d'incertitude est obtenu si +\begin{align*} +L_{\Delta f} L_f^i h = 0 & 0 \leq i \leq r-2 \\ +L_{\Delta g} L_f^i h = 0 & 0 \leq i \leq r-1 +\end{align*} + +Ce résultat ne peut être vérifié qu'a posteriori car $\Delta f$ et $\Delta g$ sont inconnues. +\end{rmq} + +\newpage +\section*{Robustesse en NL - Commande par mode glissant} + +%\imgt{8/1} + +Un terme $u_r$ est ajouté à la commande de départ $u_{eq}$ ... + +\begin{exemple}[Onduleur de tension commandé en courant] +Sans avoir à modéliser la charge, on veut imposer la forme de courant : +%\imgt{8/2} +\end{exemple} + +\subsection*{Éléments de synthèse de la commande} + +\begin{enumerate} +\item Synthétiser une commande sans prise en compte de l'incertitude ni de la perturbation : surface de glissement (poursuite asymptotique) +\item Commande gardant les états sur la surface de glissement ayant pour hypothèse l'incertitude ou la perturbation bornées : variation de la structure du système par commutation +\end{enumerate} + +\begin{dfn}[Surface de glissement ou commutation] +$S(x,t)$ est la surface autour (dans un voisinage) de laquelle le système évolue avec une dynamique imposée par $S$. +\end{dfn} + +\begin{dfn}[Système à structure variable] +Un système est à structure variable si son entrée commute entre deux valeurs suivant une logique bien spécifique $\sigma(x)$ +%\img{0.5}{8/3} +\end{dfn} + +\begin{dfn}[Commande par mode glissant] +Commande discontinue ayant pour objectif de faire converger le système en $S$. On utilise la fonction de Lyapunov \[ V(x,t) = \frac{1}{2}S^2(x,t) \] + +Pour avoir convergence vers la surface de glissement, il faut avoir +\[ \dot{V}(x,t) = S(x,t) \dot{X}(x,t) \leq 0 \] + +$\sigma(x)$ est la logique qui impose $S\dot{S} \leq 0$ +\end{dfn} + +\begin{rmq} +$S\dot{S}$ est la condition d'existence d'un régime glissant sur la surface $S$. +\end{rmq} + +\subsection*{Application de la commande par mode glissant} + +La poursuite asymptotique est une méthode de détermination de $S$. + +Soit $\epsilon(t) = y_c(t) - y(t)$ où $y_c$ est la consigne et $y$ la sortie. + +On pose $S = \epsilon^{(m)}(t) + \beta_{m-1}\epsilon^{(m-1)} + \dots + \beta_1 \dot{\epsilon} + \beta_0 \epsilon$ où $\beta_i, i =0,\dots,m-1$ sont choisis pour imposer la dynamique de convergence. + +\begin{rmq} +Par exemple, on peut choisir $S = (\frac{d}{dt} + \ld)^m \epsilon, \ld >0$ + +Choix de la commande (bouclage linéarisant) +\end{rmq} + +On pose $m=r-1$ où $r$ est le degré relatif et on a +\begin{align*} +u & = \frac{1}{L_gL_f^{r-1}h(x)} (-L_f^rh(x) + y_c^{(r)} + \sum_{i=1}^r \beta_{i-2} \epsilon^{(i-1)} + \alpha K sgn(S) ) \\ +u & = \frac{1}{L_gL_f^{r-1}h(x)} (-L_f^rh(x) + y_c^{(r)} + \dot{S} + \alpha K sgn(S) ) +\end{align*} + + +Ainsi en utilisant le changement de variable $z_i = L_f^{i-1}h(x) = \phi_i(x), i = 1,\dots,r$, la commande linéarisante avec poursuite asymptotique et robuste s'écrit : +\[ u = \frac{1}{b(z,\eta)} (-a(z,\eta) + y_c^{(r)} + \dot{S} + \alpha K sgn(S)) \] +avec pour modèle normal : +\begin{align*} +\dot{z_1} & = z_2 \\ +& \vdots \\ +\dot{z_{r-1}} & = z_r \\ +\dot{z_r} & = y_c^{(r)} + \dot{S} + \alpha K sgn(S) + \Delta a (z,\eta) \\ +\dot{\eta} & = q(z,\eta) + \Delta q(z,\eta) + \Delta p(z,\eta)u \\ +\avec & \Delta a (z,\eta) = L_{\Delta f} L_f^{r-1} h(x), \Delta q(z,\eta) = L_{\Delta f} \eta, \Delta p(z,\eta) = L_{\Delta f} \eta +\end{align*} + +On suppose que $|\Delta a (z,\eta)| < K < \infty$ donc pour avoir $\dot{z_r} = y_c^{(r)}$, on doit poser $\dot{S} = - \alpha K sgn(S) - \Delta a(z,\eta)$. + +Cas $S>0 \Rightarrow \dot{S} < -K(\alpha-1) < 0 \si \alpha > 1$ + +Cas $S<0 \Rightarrow \dot{S} > K(\alpha-1) > 0 \si \alpha < 1$ + +Ainsi on vérifie la condition d'existence du régime glissant, alors quand la trajectoire atteint $S$, alors $y \tdv y_c$ suivant la dynamique imposée par $S$. +\end{document} diff --git a/424-Systeme_Non_Lineaires/Cours/main.tex b/424-Systeme_Non_Lineaires/Cours/main.tex new file mode 100644 index 0000000..c7ad584 --- /dev/null +++ b/424-Systeme_Non_Lineaires/Cours/main.tex @@ -0,0 +1,27 @@ +\documentclass{../../cours} +\usepackage{../../raccourcis} +% Mise en page +\title{Notes de Cours} +\author{Pierre-Antoine Comby} +\teacher{Mohamed Abbas Turkis} +\module{424 \\ Commandes de système non linéaires} +\usepackage{multicol} +\begin{document} + +\maketitle +\tableofcontents +\part{Analyse de la stabilité} +\chapter{Classification} +\subfile{chap1.tex} +\chapter{Stabilité des systèmes linéaires} %premier cours en 2019 +\subfile{chap4.tex} +\chapter{Linéarisation} +\subfile{chap2.tex} +\chapter{Methode du premier harmonique} +\subfile{chap3.tex} +\chapter{Commandabilité et observabilité en non linéaire} +\subfile{chap5.tex} +\part{Outils pour la commande non linéaire} +\part{Synthèse de lois de commandes non linéaires} +%\subfile{chap2.tex} +\end{document} diff --git a/424-Systeme_Non_Lineaires/TD/TD1/1.png b/424-Systeme_Non_Lineaires/TD/TD1/1.png new file mode 100644 index 0000000000000000000000000000000000000000..f7272338fbeed21e34ef043cc2409f2270ba0122 GIT binary patch literal 85722 zcmXtAcRbbY`!`M;5|S0-&Cre?IqpUGH_B5KVQ(3#1IBczAdhl$GSP@$d+m@$m59APL|n@@r)m z;eUirWR!J~NaVz<#ti)Nva7tl>pe$H*QZz)3p^_a$Hx}DPae5gSU5bfc68k&Y?8#o zL*XgQN$YsLUGuhbdOPB_Ti3AH!RIG$q?1X7ZYHI@6(e{(ROqq62a9e!CeMYfc4zj~ z&*p!wc9(oce_2)(a)`b*mYJPt&w*6&E$B&Io0Q9GWUbhl>V@=so5W z-=%qeBSN2tJzhngZ6MSa+kry1uDDuST5`qr{Ed{nbB8-x>6Jp39Gmj#a)zc7n(8_= zX@=(OlL11UpGuC5%Y2#s=lQMIeU@nMFH76XXsURX_@J&Y0%X1AR3T`|;qTw$2CQ|8 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zO8f%?a_+E_l{QG|$g$Jw|K{c~pkXBQ+f%-`qdt$JXFJHz<=-}?1?Zo^h zhvKUG!$-nRQraH$`|>fRcy7N&p!1|K|KmTqXTkSm+Qk9O(>|NPN%y)x@j zz)Xxx0e4Sz3Rl&D^w9$)Rh9$eK33d}O;0y$H#ClE;rm$fz50Rw$be_|#M6KoeJ7Gz9sK~U$$Y6$k{J8D(RKe8n@YarJTeLL`*oghRlMd{G z4wRNZG!&~ DAP%{F literal 0 HcmV?d00001 diff --git a/424-Systeme_Non_Lineaires/TD/TD1/TD1.tex b/424-Systeme_Non_Lineaires/TD/TD1/TD1.tex new file mode 100644 index 0000000..7f2a9e9 --- /dev/null +++ b/424-Systeme_Non_Lineaires/TD/TD1/TD1.tex @@ -0,0 +1,211 @@ +\documentclass{article} +\input{../../preambule/preambule} + +\newcommand{\nom}{TD1 : Espace de phase et stabilité} +\renewcommand{\nomentete}{UE424 - \nom} + + + +\begin{document} + +\titre{\nom} +\section*{Exercice 1} +\begin{enumerate} +\item Pour trouver les points d'équilibres, on annuler les dérivées des positions: +\[ +\left \{ \begin{matrix} +0 = \sigma(x_2-x_1)\\ +0 = \rho x_1 - x_2-x_1x_3\\ +0 = -\beta x_3 + x_1 x_2 +\end{matrix} \right. \Leftrightarrow +\left \{ \begin{matrix} +x_1 = x_2\\ +x_1(\rho -1 -x_3) = 0 \\ +\beta x_3 = x_1^2 +\end{matrix} \right.\] +Si $ x_1 = 0$, alors: +\[\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} = \begin{pmatrix}0\\0\\0\end{pmatrix}\] +Si $x_1 \neq 0$, alors +\[\left \{ \begin{matrix} +x_3 = \rho -1\\ +\text{si $\rho > 1$, } x_1 = \pm \sqrt{\beta(\rho -1)} = x_2 +\end{matrix}\right. +\text{ ou alors } +\left \{ \begin{matrix} +x_3 = \rho -1\\ +\text{si $\rho < 1$, } x_1 = \pm j\sqrt{\beta(1-\rho)} = x_2 +\end{matrix}\right.\] + + +\item Donnons la linéarisation tangente du système autour du point d'équilibre en $\rho = 1$: +\[ +\left \{ \begin{matrix} +\dot{x_1} = \sigma(x_2-x_1)\\ +\dot{x_2} = x_1 - x_2-x_1x_3\\ +\dot{x_3} = -\beta x_3 + x_1 x_2 +\end{matrix} \right. \Leftrightarrow +\left \{ \begin{matrix} +\dot{x_1} = f_1(x))\\ +\dot{x_2} = f_2(x)\\ +\dot{x_3} = f_3(x) +\end{matrix} \right.\] +Ainsi, on a en linéarisant autour de 0: +\[\delta \dot{x} = \begin{pmatrix} +\left. \frac{\partial f_1}{\partial x_1}\right |_0 & \left. \frac{\partial f_1}{\partial x_2}\right |_0 & \left. \frac{\partial f_1}{\partial x_3}\right |_0 \\ \left. \frac{\partial f_2}{\partial x_1}\right |_0 & \left. \frac{\partial f_2}{\partial x_2}\right |_0 & \left. \frac{\partial f_2}{\partial x_3}\right |_0 \\ +\left. \frac{\partial f_3}{\partial x_1}\right |_0 &\left. \frac{\partial f_3}{\partial x_2}\right |_0 & \left. \frac{\partial f_3}{\partial x_3}\right |_0 +\end{pmatrix} \delta x \Rightarrow +A = \begin{pmatrix} +-\sigma & \sigma & 0 \\ 1 & -1 &0 \\ 0& 0& -\beta +\end{pmatrix}\] + +Stabilité en linéaire de $x_0 = \begin{pmatrix}0\\0\\0\end{pmatrix}$ +On calcule $det(\lambda I - A) = 0$, ie: +\begin{align*} +det\begin{pmatrix}\lambda + \sigma & -\sigma & 0 \\ -1 &\lambda +1 &0\\ 0 & 0 &\lambda + \beta\end{pmatrix} &= (\lambda + \beta)((\lambda + \sigma ) (\lambda + 1 ) - \sigma)\\ +&= (\lambda + \beta)(\lambda^2 + (1+\sigma)\lambda +\intertext{On a donc les trois équations suivantes:} +\lambda &= - \beta\\ +\lambda &= 0\\ +\lambda &= -(\sigma+1) +\end{align*} + +Il n'existe pas de point d'équilibre en linéaire ce qui contredit le résultat en N.L où nous avons pour seul point d'équilibre $ x = \begin{pmatrix} +0\\0\\0 +\end{pmatrix}$. + +\end{enumerate} + +\section*{Exercice 2 : Asservissement à relais} + +On considère le système constitué d'un moteur a courant continu asservie en position avec une correction tachymétrique, donné par la figure ci-dessous. R(.) représente la caractéristique d'un relais symétrique avec seuil $\Delta$ et hystérésis $h$. + +\begin{center} +%\includegraphics[scale=0.5]{figure1.png} +\end{center} + +\begin{enumerate} +\item On pose $e(t) = 0$, la transformée inverse donne donc, d'une part: +\begin{align*} +\omega &= \frac{d\theta}{dt} +\intertext{Ce qui conduit à: } +\omega(t) + \tau \frac{d\omega(t)}{dt} &= \frac{d\theta(t)}{dt} + \tau \frac{d^2\theta(t)}{dt^2}\\ +&= K R(-L\frac{d\theta(t)}{dt} - \theta(t))\\ +&= -KR(L\frac{d\theta(t)}{dt} + \theta(t)) +\intertext{avec comme condition initiale:} +\theta(t=0) &= 0\\ +\omega(t=0) &= 0 +\end{align*} + +Si l'on pose maintenant $e = e_0u(t)$, on a simplement: +\[\frac{d\theta(t)}{dt} + \tau \frac{d^2\theta(t)}{dt^2} = -KR(L\frac{d\theta(t)}{dt} + \theta(t) - e)\] +avec comme condition initiale $\theta(0) = -e_0$ et $\dot{\theta} = \omega_0 = 0$.\\ + +Remarque : si on pose $ \theta ' = \theta - e_0$ on retrouve la même équation différentielle et cela n'influence pas la transformée de Laplace, les deux sont donc équivalent. + +\item On a pour le temps réduit $ \overline{t} = \frac{t}{\tau}$.\\ +Attention! La fonction R(.) fait sortir un $U_0$ a ne pas oublier.\\ + Et on pose en identifiant après avoir injecté le $\tau$ provenant de la normalisation du temps, $\overline{\theta} = \frac{\theta}{KU_0\tau}$.\\ + On trouve alors simplement $\overline{\omega} = \frac{d\overline{\theta}}{d\overline{t}} = \frac{\omega}{KU_0}$. + Reste plus qu'à identifier les constantes:\\ + On a simplement $\beta = \frac{L}{\tau}$.\\ + Et d'une façon presque obscure $a = \frac{\Delta + h}{2KU_0\tau}$ et $\alpha a = \frac{\Delta - h}{2KU_0\tau}$. + + \begin{center} + %\includegraphics[scale=0.5]{figure2.png} + \end{center} + +\item On pose $\frac{d\overline{\theta}}{d\overline{t}} + \frac{d^2\overline{\theta}}{d\overline{t}^2} = \lambda$ avec $\lambda = 1$ , 0 ou -1 en fonction de $\epsilon$ ou de $\frac{d\epsilon}{dt}$.\\ + +On a comme condition initiale : $\overline{\theta}(t=0) = \overline{\theta_0}$ et $\overline{\omega}(t=0) = \overline{\omega_0}$. Ce qui conduit à: +\begin{align*} +\overline{\omega}(t) + \frac{d\overline{\omega}}{d\overline{t}} = \lambda &\Rightarrow \overline{\omega}(t) = \overline{\omega_0}e^{-\overline{t}} + \lambda(1 - e^{-\overline{t}}) +\intertext{On a donc en variable $\overline{\theta}$} +\frac{d\overline{\theta}}{d\overline{t}} &= \overline{\omega_0}e^{-\overline{t}} + \lambda(1 - e^{-\overline{t}}) +\intertext{d'où:} +\overline{\theta}(t) - \overline{\theta_0} &= \overline{\omega_0} - \overline{\omega_0}e^{-\overline{t}} + \lambda \overline{t} - \lambda(1 - e^{-\overline{t}}) +\end{align*} + +\item Pour décrire l'espace des phases, on pose $x_1 = \overline{\theta}$ et $x_2 = \overline{\omega}$.Par élimination de $\overline{t}$, en utilisant la méthode explicite on a: + +\begin{align*} +x_2(t) + x_1(t) &= \overline{\omega_0} + \overline{\theta_0} + \lambda\overline{t} +\intertext{d'où:} +x_2 - \lambda &= (\overline{\omega_0} - \lambda)e^{-\overline{t}} +\intertext{et ainsi:} +\overline{t} &= ln\left(\frac{\overline{\omega_0}-\lambda}{x_2-\lambda}\right) +\intertext{ainsi, en remplaçant de façon explicite le temps réduit:} +x_2(t) + x_1(t) &= \overline{\omega_0} + \overline{\theta_0} + \lambda ln\left(\frac{\overline{\omega_0}-\lambda}{x_2-\lambda}\right) +\end{align*} + +Avec la méthode implicite on a $\frac{dx_2}{d\overline{t}} = \lambda -x_2$ et $\frac{dx_1}{d\overline{t}} = x_2$. Ainsi on a $\frac{dx_2}{dx_1} = \frac{\lambda-x_2}{x_2}$ et en intégrant, on retrouve le résultat précédent. + +\item L'allure de l'espace de phase dépend de la valeur de $\lambda$.\\ + +Pour $\lambda = 0$ on a directement $x_1 + x_2 = \overline{\theta_0} + \overline{\omega_0}$ + +Pour $\lambda $ on a + + +Comportement asymptotique +\[\overline{t} \leftarrow \infty \Rightarrow \left \{ \begin{matrix} +x_1 = \pm \infty \\ +x_2 = \lambda +\end{matrix} \right. \] +\[\overline{t} \leftarrow -\infty \Rightarrow \left \{ \begin{matrix} +x_1 \approx -\overline{\omega_0} e^{-\overline{t}}+\lambda e^{-\overline{t}} \\ +x_2 \approx \overline{\omega_0} e^{-\overline{t}}-\lambda e^{-\overline{t}} +\end{matrix} \right. \] +On a dans le deuxième cas: +$x_1(t) = - x_2(t)$ et $\frac{dx_2}{dx_1} = \frac{\lambda - x_2}{x_2} =_{x_2=0} + \infty$ + +\begin{align*} +\frac{d^2x_2}{dx_1^2} &= \frac{-\lambda(\lambda-x_2)}{x_2^3} +\intertext{Selon la valeur de $\lambda$ on a plusieurs solutions:} +\lambda = -1 \text{ et, } x_2 >-1 &\Rightarrow \text{concavité tournée vers $x_1 < 0$}\\ +\lambda = 1 \text{ et, } x_2 < 1 &\Rightarrow \text{concavité tournée vers $x_1 > 0$}\\ +\end{align*} + +\img{0.25}{1.png} + +\newpage +On a en sortie du comparateur: $\epsilon = x_1 + \beta x_2$ +Sachant que l'on a la caractéristique: (attention, on a permuté avec $-R(\epsilon)$ + +\img{0.4}{2.png} + +On en déduit que: + +\img{0.4}{3.png} + +Ainsi, selon la où l'on est, on va avoir différent $\lambda$, et on va pouvoir recouper ce graph avec celui de l'espace de phase précédent pour avoir le comportement du système dans l'espace de phase. +On parcourt donc l'espace de phase en partant du point P, puis on se déplace vers le point Q par la droite de pente -1, puis de Q a R et S pour revenir vers T sur la portion de courbe ou se situe P. + +\img{0.4}{4.png} + +Si $x_{2T} < x_{2P}$, alors on a stabilité et on converge vers le point d'équilibre 0.\\ +Si $x_{2T} > x_{2P}$, alors on a un comportement instable et le système diverge.\\ +Si $x_{2T} = x_{2P}$, alors on est sur le cycle limite.\\ + +\item On étudie chaque portion du cycle. + +Entre le point P et Q on a comme relation: +\begin{align*} +x_{1p} + \beta x_{2p} & = -\alpha a\\ +x_{1q} + \beta x_{2q} & = a +x_{1p} + x_{2p} & = x_{1q} + x_{2q} \text{ (dynamique)} +\end{align*} + +Entre le point Q et R on a comme relation: +\begin{align*} +x_{1q} + \beta x_{2q} & = a\\ +x_{1r} + \beta x_{2r} & = \alpha a\\ +x_{1r} + x_{2r} & = x_{1q} + x_{2q} - ln\left(\frac{1+x_{2q}}{1+x_{2r}}\right) \text{ (dynamique)} +\end{align*} + +Par symétrie, l'obtention du cycle limite vérifie $x_{2r} = -x_{2p}$. On a donc donc 6 inconnu et 6 équations différentes. + +Ainsi, pour imposer le comportement du système, on fixe un cycle limite et l'on impose les valeurs de $a$ et $\alpha$ pour l'obtenir. +\end{enumerate} + + +\end{document} \ No newline at end of file diff --git a/424-Systeme_Non_Lineaires/TD/TD2/2.png b/424-Systeme_Non_Lineaires/TD/TD2/2.png new file mode 100644 index 0000000000000000000000000000000000000000..d6e94935a9c957cc2fea805cda5a48e6437e71a7 GIT binary patch literal 38481 zcmXtAcRZEv`$xw}#)*?oKH`Kpq!Qwgy|Q=q-ehmG6(KT0$X?0H%3evbLXu>!?46MH zyL?}--}$28Or@Yic@Qrd3nj+SnoCN36uRt}E#7Pno^Tr4acT&*45HZQaYI%zkT3^{N>hCv=QVy2bx(NqlTsbK^ADv* zA3Pr4GE!^MOYQclsYo!N`~KPR!O!h0X8(~9nTQ68z9IN@Lua4AUOJ`H%E#y5zW}f9 z-4OyB@m@Fiw*d@TWRCZg|B+wDs$cq4NJRE?tc>u#fB$6Ss$B;tMIN7x{KRy_Jq2)FjhxS{WhsLfY(jlQa$=CDmtH`M+kgd*il7Z4Wv0fCJyXM{=swkbrM@7RGjek_Il5x5EcZI0J z-)gWg_v0`q(&as$YF(j`s%Q)eU8iElaJz(Su+D!5>Zi6B*E`F zRy`V|R1x-w(AVa~X6EKkG0(q$*I2HOp4u$HvU78*myO5^^6aF|z>{1cwR<;g3-1!S 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+\img{0.5}{2.png} + +On pose \[ N(X) = \frac{P+jQ}{X} \avec Q=0 \] +\begin{align*} +P &= \frac{4\omega}{\pi} \int_{t_1}^{\frac{\pi}{2\omega}}(Xsin(\omega t) - \frac{\Delta}{2})sin(\omega t) dt\\ +&= \frac{4\omega}{\pi} \int_{t_1}^{\frac{\pi}{2\omega}}(Xsin^2(\omega t) - \frac{\Delta}{2}sin(\omega t)) dt\\ +&=-\frac{2\Delta}{\pi}cos(\omega t_1) - \frac{4 X \omega}{\pi}(\frac{t_1}{2} - \frac{\pi}{4\omega} - \frac{sin(2t_1 \omega}{4\omega})\\ +\text{Or, } & Xsin(\omega t_1) = \frac{\Delta}{2} \Rightarrow t_1 = \frac{1}{\omega} arcsin(\frac{\Delta}{2X})\\ +&\Rightarrow P = X (1 - \frac{2}{\pi}(arcsin(\frac{\Delta}{2X}) + \frac{\Delta}{2X}\sqrt{1-(\frac{\Delta}{2X})^2} +\end{align*} + +On a alors $N(X) = \frac{P}{X}$. + + +\item Pour le relais avec hystérésis + +\img{0.5}{3} +On pose $X\sin(\omega t_1) = \frac{h}{2} \donc t_1 = \frac{1}{\omega} arcsin(\frac{h}{2X})$ + +\begin{figure}[h!] +\centering +\includegraphics[scale=0.5]{4} +\includegraphics[scale=0.5]{5} +\end{figure} + +\begin{multicols}{2} +\begin{align*} +P & = \frac{\omega}{\pi} \int_{[T]} y(t) \sin(\omega t) dt \\ +& = 2\frac{\omega}{\pi} \int_{t_1}^{\frac{\pi}{\omega}+t_1} M \sin(\omega t) dt\\ +& = 2\frac{\omega}{\pi}M . \frac{-\cos(\pi+\omega t_1) + \cos(\omega t_1)}{\omega} \\ +P & = \frac{4M}{\pi} \sqrt{1-(\frac{h}{2X})^2} +\end{align*} + +\begin{align*} +Q & = \frac{\omega}{\pi} \int_{[T]} y(t) \cos(\omega t) dt \\ +& = 2\frac{\omega}{\pi} \int_{t_1}^{\frac{\pi}{\omega}+t_1} M \cos(\omega t) dt \\ +& = 2\frac{\omega}{\pi}M.\frac{\sin(\pi + \omega t_1) - \sin(\omega t_1)}{\omega} \\ +Q & = -\frac{4M}{\pi} \frac{h}{2X} +\end{align*} +\end{multicols} + +\[ |N(X)| = \frac{\sqrt{P^2 + Q^2}}{X} = \frac{4M}{\pi X} \] + +\item Pour le jeux sans inertie aval +%\img{0.5}{6} + +\[ y(t) = +\left\{ +\begin{array}{cc} +x(t) - \alpha & \text{ sur } DA\\ +M & \text{ sur } AB \\ +x(t) + \alpha & \text{ sur } BC \\ +-M & \text{ sur } CD +\end{array} +\right. +\] + +\[ |N(X)| = \frac{1}{2\pi}\sqrt{ (1+\cos2\beta)^2 - (\pi+2\beta+\sin2\beta)^2} \avec \beta = \arcsin(1-2\frac{\alpha}{M}) \] + +\end{enumerate} + +\section*{Exercice II : Asservissement avec boucle secondaire} + +\begin{enumerate}\setlength{\itemsep}{1cm} + +\item Relations entre les différentes variables +\begin{align*} +s(p(1+T_2p)) = k_2 w & \Rightarrow \dd{s}{t} + T_2 \dd{^2 s}{t^2} = k_2 w(t) \\ +w = \phi(x) & \\ +x = r - ks & \\ +r(1+T_1 p) = -k_1 s & \Rightarrow r(t) + T_1 \dd{r}{t} = -k_1 s(t) +\end{align*} + +\item On exprime $x$ en fonction de $s$ et $s$ en fonction de $x$ : +\begin{align*} +x & = \frac{-k_1}{1+T_1 p} s - ks \\ +s & = \frac{k_2}{p(1+T_2p)}\phi(x) +\end{align*} +donc +\[ p(1+T_2p)(1+T_1p)x = -(k_1+k(1+T_1p))k_2\phi(x) \] +\[ T_2T_1x^{(3)} + (T_1+T_2)x^{(2)} + x^{(1)} = -k_2(k_1+k)\phi(x)-kk_2T_1 \dd{\phi(x)}{t} \] + +\item On peut appliquer l'approximation du 1er harmonique à la NL car : +\begin{itemize} +\item La NL est statique +\item Un filtre passe-bas d'ordre relatif $>1$ est en aval de la NL + +\begin{rmq} +Si le filtre est $\frac{p+1}{p^2 / \omega^2 + 2m/\omega p +1}$, on ne peut pas appliquer la méthode du 1er harmonique. +\end{rmq} +\end{itemize} + +\item On a $Q=0$ car la NL est impaire. +\[ P = \frac{2}{T} \int_0^T y(t) \sin(\omega t) dt = \frac{2}{T} ( \int_0^{T/2} M\sin (\omega t) dt - \int_{T/2}^T M\sin(\omega t) dt) = \frac{4M}{\pi} \] + +\item Avec $x(t)=X\sin(\omega t)$, on a donc l'approximation $w(t)=N(X).x(t)$ avec $N(x)=\frac{4M}{\pi X}$. + +\item Analyse harmonique en remplaçant $\frac{d}{dt}=j\omega$ +\[ -T_1T_2 j\omega^3 - (T_1+T_2) \omega^2 + j(1+kk_2T_1N(X)) \omega + (k_1+k)k_2N(X) = 0 \] + +On en déduit les équations algébriques du cycle limite en prenant parties réelle et imaginaire : +\begin{align*} +(k_1+k)k_2 N(X) - (T_1+T_2) \omega^2 & = 0 \\ +(1+kk_2T_1N(x))\omega - T_1T_2 \omega^3 & =0 +\end{align*} + +\item Cycle limite.\\ + +\paragraph{Existence du cycle limite} On cherche une solution aux équations algébriques : +\begin{align*} +Re=0 & \Rightarrow \omega_0^2 = \frac{(k_1+k)k_2N(x)}{T_1+T_2} \\ +Im=0 & \Rightarrow X_0 = \frac{4Mk_2T_1(k_1T_2-kT_1)}{\pi(T_1+T_2)} +\end{align*} + +On peut alors réécrire \[\omega_0^2 = \frac{k_1+k}{T_1(k_1T_2-kT_1)} \] + +\paragraph{Stabilité du cycle limite} +\[ \drond{R}{X}|_0 \drond{I}{\omega}|_0 - \drond{I}{X}|_0 \drond{R}{\omega}|_0 > 0 \] + +\begin{align*} +\drond{R}{X}|_0 & = (k+k_1)k_2 \dd{N}{X}|_0 < 0 \\ +\drond{R}{\omega}|_0 & = -2(T_1+T_2)\omega_0 < 0 \\ +\drond{I}{X}|_0 & = kk_2T_1 \dd{N}{X}|_0 \omega_0< 0 \\ +\drond{I}{\omega}|_0 & = 1 + kk_2T_1N(X_0)-3T_1T_2\omega_0 = -2T_1T_2\omega_0^2 < 0 +\end{align*} + +Le cycle limite est stable si $\drond{R}{X}|_0 \drond{I}{\omega}|_0 - \drond{I}{X}|_0 \drond{R}{\omega}|_0 > 0$ +\[-2(k+k_1)k_2\dd{N}{X}T_1T_2\omega_0^2 + 2(T_1+T_2)\omega_0^2kk_2T_1\dd{N}{X}> 0\] + +soit +\[T_2(k+k_1)\dd{N}{X}|_0 + (T_1+T_2)k\dd{N}{X}|_0 > 0\] + +soit \[T_1k-T_2k_1<0\] + +Même condition que celle d'existence du cycle limite. + +%\img{0.5}{7} +\end{enumerate} + + +\section*{Exercice III : Contre-exemple} + +\begin{enumerate}\setlength{\itemsep}{1cm} + +\item Voir Exercice I : +\[ N(x) = \frac{4M}{\pi X}\sqrt{1-(\frac{h}{2X})^2} - j\frac{2Mh}{\pi X^2} = N_P(x) + jN_Q(X) \quad \et \quad |N(X)| = \frac{4M}{\pi X}\] + +Le lieu critique est défini par +\[ - \frac{1}{N(x)} = \frac{-N_P(X) + j N_Q(X)}{|N(X)|^2} = -\frac{\pi X}{4M}\sqrt{1-(\frac{h}{2X})^2} - j\frac{\pi^2h}{8M} \] + +On trace le lieu de Nyquist de la fonction de transfert de $\frac{K}{1+\tau p}$ ainsi que celui de $-\frac{1}{N(X)}$ : + +%\img{0.5}{8} + +Il n'y a pas d'intersection entre les deux : d'après la méthode du 1er harmonique, il n'y a donc pas de cycle limite. + +\item $KM > h/2$ + +L'entrée du filtre du 1er ordre $u=M$ + +\[y(t) = KM(1-e^{-t/\tau}) \Rightarrow \exists t_1 \tq y(t_1) > h/2 \car KM>h/2 \] +\[x(t) = -y(t) < -h/2 \Rightarrow u = -M\] + +et avec le même raisonnement, +\[ \exists t_2 \tq y(t_2) < -h/2 \] +\[ x(t) > h/2 \] + +donc pour $KM>h/2$, il existe un cycle limite. + +\item On obtient une contradiction car le filtre est de degré relatif égal à 1. +\end{enumerate} + +\end{document} diff --git a/424-Systeme_Non_Lineaires/TD/TD3a/TD3a.tex b/424-Systeme_Non_Lineaires/TD/TD3a/TD3a.tex new file mode 100644 index 0000000..4af6bf5 --- /dev/null +++ b/424-Systeme_Non_Lineaires/TD/TD3a/TD3a.tex @@ -0,0 +1,90 @@ +\documentclass{article} +\input{../../preambule/preambule} + +\newcommand{\nom}{TD3 : Stabilité en non linéaire} +\renewcommand{\nomentete}{UE424 - \nom} + + + +\begin{document} + +\titre{\nom} + +\section*{Exercice I : Stabilité au sens de Lagrange et de Lyapunov} + +\begin{enumerate}\setlength{\itemsep}{1cm} + +%---------------- Question 1 +\item \begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}\setlength{\itemsep}{8mm} + +\item On défini le système par \[\dot{x}=(6t\sin t - 2t)x\] donc +\begin{align*} +\frac{dx}{x} & = (6t \sin t - 2t) dt \\ +[\ln(u)]_{x_0}^x & = 6( [-\tau \cos \tau ]_{t_0}^t + \int_{t_0}^t \cos \tau d \tau - [\tau^2]_{t_0}^t \\ +\ln(x) - \ln(x_0) & = -6t\cos t + 6t_0\cos t_0 + 6\sin t - 6\sin t_0 - t^2 + t_0^2 +\end{align*} + +Donc on a la trajectoire : +\[ x(t) = x_0 \exp(-6t\cos t + 6t_0\cos t_0 + 6\sin t - 6\sin t_0 - t^2 + t_0^2) \] + +\item Stabilité au sens de Lyapunov : +\[ \forall \epsilon > 0, \exists \delta >0 \tq ||x_0|| \leq \delta \Rightarrow ||x|| \leq \epsilon \] + +Soit $\epsilon>0 \tq |x(t)| \leq \epsilon$. Exprimons $\delta$ en fonction de $\epsilon$ tel que $|x_0| \leq \delta$. +\begin{align*} +|x(t)| & \leq |x_0| \exp (6t_0 \cos t_0 - 6\sin t_0 + t_0^2 + 6 + 6 t - t^2) \\ +\text{Or, } & 0 < (3-t)^2 = 9-6t+t^2 \Rightarrow 6t-t^2 < 9 \\ +\donc |x(t)| & \leq |x_0| C \quad \avec C = \exp(6t_0 \cos t_0 - 6\sin t_0 + t_0^2 + 12) > 0 \\ +\Rightarrow \delta & = \frac{\epsilon}{C} +\end{align*} + +\begin{rmq} +Le fait que le $\delta$ dépende de $t_0$ n'empêche pas que l'origine soit stable au sens de Lyapunov. Cela montre que la stabilité n'est pas uniforme. +\end{rmq} + +\item Stabilité au sens de Lagrange : +\[ \forall \delta > 0, \exists \epsilon >0 \tq |x_0| \leq \delta \Rightarrow |x| \leq \epsilon \] + +$t_0=2\pi n$ et $t=t_0 + \pi$ + +\begin{align*} +x(t) & = x_0 \exp(6.2\pi n + A\pi^2 n^2 - 6(2\pi n + \pi) - (2\pi n + \pi)^2) \\ +& = x_0\exp((4n+1)\pi(6-\pi)) \tdv \infty \si n \tdv \infty +\end{align*} + +$|x_0| \leq \delta$ alors que $\nexists \epsilon > 0 \tq |x| \leq \epsilon$ : l'origine n'est pas stable au sens de Lagrange. + + +\end{enumerate} + +% --------------- Question 2 +\item \begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}\setlength{\itemsep}{8mm} +\item On définit le système par +\[ \dot{x} = -\frac{x}{t+1} \] + +\begin{align*} +\frac{dx}{x} & = - \frac{dt}{t+1} \\ +\ln(\frac{x}{x_0}) & = \ln(\frac{1+t_0}{1+t}) \\ +\end{align*} + +Donc on a la trajectoire : +\[ x(t) = x_0 \frac{1+t_0}{1+t} \] + +\item Stabilité au sens de Lagrange : +\[ \forall \delta > 0, \exists \epsilon >0 \tq |x_0| \leq \delta \Rightarrow |x| \leq \epsilon \] + +Soit $\delta > 0 \tq |x_0| \leq \delta$, il faut exprimer $\epsilon$ en fonction de $\delta \tq |x(t)| \leq \epsilon$. + +\[ |x(t)| = |x_0|\frac{1+t_0}{1+t} \avec t,t_0>0 \et t\geq t_0 \donc \frac{1+t_0}{1+t} \leq 1\] + +On prend $\epsilon=\delta$ et l'origine est stable au sens de Lagrange. + +\item Stabilité au sens de Lyapunov : +\[ \forall \epsilon > 0, \exists \delta >0 \tq |x_0| \leq \delta \Rightarrow |x| \leq \epsilon \] + +Soit $\epsilon>0$. On pose $\delta=\epsilon$ +\[ |x_0| \leq \delta=\epsilon \Rightarrow |x| = |x_0|\frac{1+t_0}{1+t} \leq \epsilon \frac{1+t_0}{1+t} \leq \epsilon \] chibrage de l'exo +\end{enumerate} +\end{enumerate} + +\end{document} \ No newline at end of file diff --git a/424-Systeme_Non_Lineaires/TD/TD3b/TD3b.tex b/424-Systeme_Non_Lineaires/TD/TD3b/TD3b.tex new file mode 100644 index 0000000..4a940cd --- /dev/null +++ b/424-Systeme_Non_Lineaires/TD/TD3b/TD3b.tex @@ -0,0 +1,179 @@ +\documentclass{article} +\input{../../preambule/preambule} + +\newcommand{\nom}{TD3 : Stabilité et non linéarité} +\renewcommand{\nomentete}{UE424 - \nom} + + + +\begin{document} + +\titre{\nom} +\section*{Exercice 1: Stabilité en non linéaire} +\begin{enumerate} +\item + +LE bout de l'autre génie. + + +\[x(t) = x_0 \frac{1+t_0}{1+t}\] +\[\forall \epsilon > 0 \exists \delta > 0 \tq ||x_0|| \leq \delta \Rightarrow ||x(t) || \leq \epsilon\] +contraposée +\[\exists \epsilon \tq \forall \delta >0, ||x_0|| \leq \delta \et ||x(t) || \geq \epsilon\] +\begin{align*} +|x(t)| = |x_0| \frac{1+t_0}{1+t} &> \frac{|x_0|t_0}{1+t}\\ +& > |x_0| = \epsilon \si t_0 \rightarrow \infty +\end{align*} + +\end{enumerate} + + +\section*{Exercice 2: Pendule simple} + +\begin{enumerate} +\item +\begin{enumerate} +\item On applique le PFD selon l'axe $u_\theta$ :\\ +\begin{align*} +m\ddot{\theta}l &= -mg.sin(\theta)\\ +\ddot{\theta} &= -\frac{g}{l} sin(\theta) +\end{align*} + +\item On pose le vecteur $x = \begin{pmatrix}x_1\\x_2\end{pmatrix} = \begin{pmatrix}\theta \\ \dot{\theta}\end{pmatrix}$, on a donc le système d'état:\\ +\[ \dot{x} = \begin{pmatrix}\dot{x_1} \\ \dot{x_2}\end{pmatrix}\ = \begin{pmatrix}x_2 \\ -\frac{g}{l} sin(x_1)\end{pmatrix}\]\\ + +\item On a $E = \frac{1}{2}m v^2$ avec $v= l\dot{\theta}$ : +\[\boxed{E = \frac{ml^2}{2} x_2^2}\] +Et $P = mgl(1-cos(\theta))$: +\[\boxed{P = mgl(1-cos(x_1))}\] + +\item On pose $V(x) = E + P$ d'où en zéro: +\begin{align*} +V(0) &= \frac{ml^2}{2} x_{20}^2 + mgl(1-cos(x_{10})) = 0 +\intertext{si $x \neq 0$ et $|x_1| < 2 \pi$, alors $cos(x_1)<1$ donc $E+P >0$} +\frac{\partial V}{\partial x_1} &= mgl.sin(x_1)\\ +\frac{\partial V}{\partial x_2} &= ml^2 x_2\\ +\dot{V}(x) &= \dot{x_1} mglsin(x_1) + ml^2x_2\dot{x_2}\\ +\dot{V}(x) &= \dot{x_1} mglsin(x_1) + ml^2x_2(-\frac{g}{l}sin(x_1)) = 0\\ +\end{align*} +Pour $|x_1| < \pi$, l'origine sera stable + +Il n'existe pas de Q(x) tel que $\dot{V(x) \leq -Q(x)}$\\ +Barhashin : $\dot{V}(x) = \{ x_2 \in \R \et |x_1| < 2\pi \}$ \\ +Pas de stabilité asymptotique + +\end{enumerate} +\item +\begin{enumerate} + +\item On applique le PFD selon l'axe $u_\theta$ :\\ +\begin{align*} +m\ddot{\theta}l &= -mg.sin(\theta) - \alpha l \dot{\theta}\\ +\ddot{\theta} &= -\frac{g}{l} sin(\theta) -\alpha \frac{1}{m}\dot{\theta} +\end{align*} + +\item On pose le vecteur $x = \begin{pmatrix}x_1\\x_2\end{pmatrix} = \begin{pmatrix}\theta \\ \dot{\theta}\end{pmatrix}$, on a donc le système d'état:\\ +\[ \dot{x} = \begin{pmatrix}\dot{x_1} \\ \dot{x_2}\end{pmatrix}\ = \begin{pmatrix}x_2 \\ -\frac{g}{l} sin(x_1)- \frac{\alpha}{m}x_2\end{pmatrix}\]\\ + +\item On a $E = \frac{1}{2}m v^2$ avec $v= l\dot{\theta}$ : +\[\boxed{E = \frac{ml^2}{2} x_2^2}\] +Et $P = mgl(1-cos(\theta))$: +\[\boxed{P = mgl(1-cos(x_1))}\] + +\item On pose $V(x) = E + P$ d'où en zéro: +\begin{align*} +V(0) &= \frac{ml^2}{2} x_{20}^2 + mgl(1-cos(x_{10})) = 0 +\intertext{si $x \neq 0$ et $|x_1| < 2 \pi$, alors $cos(x_1)<1$ donc $E+P >0$} +\frac{\partial V}{\partial x_1} &= mgl.sin(x_1)\\ +\frac{\partial V}{\partial x_2} &= ml^2 x_2\\ +\dot{V}(x) &= \dot{x_1} mglsin(x_1) + ml^2x_2\dot{x_2}\\ +\dot{V}(x) &= \dot{x_1} mglsin(x_1) + ml^2x_2(-\frac{g}{l}sin(x_1))-\frac{\alpha}{m}x_2 - l^2\alpha x_2^2 \leq 0\\ +\intertext{Pour $|x_1| < \pi$, l'origine sera stable} +\end{align*} +Il n'existe pas de Q(x) tel que $\dot{V(x) \leq -Q(x)}$\\ +Barhashin : +\begin{align*} +\dot{V(x)} = 0 \Rightarrow x_2 =0 \Rightarrow \dot{x_1} = 0\\ +\pour |x_1- < \pi \Rightarrow x_2 = 0\\ +\donc \dot{x_2} = 0 \Rightarrow sin(x_1) = 0 +\end{align*} +Si $x_1 = 0$, $\pi$ , $-\pi$ il n'y a pas de stabilité asymptotique.\\ + +L'origine est stable asymptotiquement pour $(x_1,x_2) \ in ]-\pi ; \pi[\times \R$ +\[ +V(x) = \begin{pmatrix} +x_1 & x_2\end{pmatrix}. \begin{pmatrix} +\frac{\alpha^2}{2m^2} & \frac{\alpha}{2m} \\ \frac{\alpha}{2m} & 1 \end{pmatrix} \begin{pmatrix} +x_1 \\x_2 +\end{pmatrix} + \frac{g}{l}(1-cos(x_1))\] +\[V(0) = 0\] +\[V(x) = \begin{pmatrix} +x_1 & x_2\end{pmatrix} P \begin{pmatrix} +x_1 \\ x_2 +\end{pmatrix} + \frac{g}{l}(1 - cos(x_1))\] +Or, $P >0$ car: +\[\frac{\alpha^2}{2m^2}>0 \et \frac{\alpha^2}{2m^2} - \frac{\alpha}{4m^2}>0 \et |x_1| < 0\pi\] +donc: + +\[P \begin{pmatrix} +P_{11} & P_{12} \\ +P_{12}^T & P_{22} +\end{pmatrix} >0 \Leftrightarrow (Lemme de Schur) P_{11} >0 \et (peutetre) P_{11} - P_{12}P_{22}^+P_{12}^T + >0 (P^+ est la pseudo inverse) +\] +\begin{align*} +\dot{V}(x) &= -\frac{1}{2}(\frac{g\alpha}{lm})x_1 sin(x_1) - \frac{\alpha}{2m} x_2^2 \leq 0\\ +&\leq -\frac{g\alpha}{4lm} x_1sin(x_1) - \frac{\alpha}{4m}x_2^2 = Q(x) +\end{align*} + +L'origine est localement asymptotiquement stable. +\end{enumerate} + + +\section*{Exercice 3: Exemple de systèmes} + +\begin{enumerate} +\item \begin{enumerate} +On pose $V(x) = \frac{1}{2}(x_1^2 + x_2^2)$, et on a bien $V(0) = 0$. +$\dot{V}(x) = x_1\dot{x_1} + x_2 \dot{x_2} = -(x_1^2 + x_2^2)$.\\ +On a alors $V(x) \leq -Q(x)$ avec $Q(x) = \frac{x_1^2 + x_2^2}{2}$, donc l'origine est globalement asymptotiquement stable.\\ + +\item Vérifions la stabilité exponentielle: +$\exists \alpha>0$, $\beta>0$, $\gamma>0$, et $x>1$ tel que $\dot{V} \leq - \gamma||x--^c$ et $\alpha||x||^c \leq V(x) \leq \beta ||x||^c$\\ +Pour la norme euclidienne, on prend $c=2$ et avec $\gamma = 1$\\ +$\dot{V} \leq -\gamma||x||^2$\\ +Avec $\alpha = \frac{1}{4}$ et $\beta = 1$ la condition 2 est respectée donc on a la stabilité exponentielle. +\end{enumerate} + +\item On considère le système suivant: +\[ \acc{\dot{x_1} = x_2 + x_3^4}{\dot{x_2} = -5sin(x_1) - x_2 + u_1}{\dot{x_3} = -kx_3 + u_2} \] +\begin{enumerate} +\item Pour $u_1 = u_2 = 0$, l'origine est stable car $\dot{x} = 0 $.\\ + +\item Pour linéariser, on passe par la matrice du jacobien prise en (0,0,0): +\[ +\begin{pmatrix} +\delta \dot{x_1} \\\delta \dot{x_2} \\\delta \dot{x_3} +\end{pmatrix} = \begin{pmatrix} +0 & 1& 0\\-5 & -1 & 0\\0 & 0& -k +\end{pmatrix}.\begin{pmatrix} +\delta x_1 \\ \delta x_2 \\ \delta x_3 +\end{pmatrix} = A \begin{pmatrix} +\delta x_1 \\ \delta x_2 \\ \delta x_3 +\end{pmatrix}\] +\[ +det(\lambda I - A) = (\lambda + k)(\lambda^2 + \lambda + 5) +\] +La stabilité suivant le critère de Routh donne que c'est stable si $k>0$ + +\item \[V(x) = 5(1-cos(x_1)) + \frac{1}{2} \dot{x_2}^2 + \frac{3}{2k}x_3^4 +|x_1| < 2\pi\] +\[\dot{V}(x) = 5x_3^4sinx_1 - x_2^2 \leq -x_3^4 - x_2^2 \leq 0 \text{(ne marche pas car ne dépend pas de $x_1$)}\] +\[\dot{V}(x) \leq \frac{5}{2} x_3^4 sin x_1 -\frac{1}{2}x_2^2 - 3 x_3^4 \leq -\frac{1}{2}x_3^4 - \frac{1}{2}x_2^2\] +\[Q(x) = -\frac{5}{2}x_3^4sinx_1 + \frac{1}{2}x_2^2 + 3x_3^4 \geq 0 \text{ a condition que } |x_1| < \pi +\] +\end{enumerate} + +\end{enumerate} +\end{enumerate} +\end{document} \ No newline at end of file diff --git a/424-Systeme_Non_Lineaires/TD/TD4/1.png b/424-Systeme_Non_Lineaires/TD/TD4/1.png new file mode 100644 index 0000000000000000000000000000000000000000..20b4328fc282fc425693c4cf4ba564f9178b8b76 GIT binary patch literal 39057 zcmXtAcRZE-`?r%0PLhN;>ENUjk|dibdnYR#d+#L4%n?FXvNu^-AxY?vag30ikZh9d ztl#DNzJC6Bo|NOhKlkTzUGH_>p&Dv;X=zw!NJvO%6&2*PNJz+Dl8}&|rY3{$-0mlt zga44b%PQ(nQ&W#ksZYWm&wI%0d1yOZc|5|oS&~>gIXhZjbvJjjv~+T}arRgxZ;&J* zL6IoR$>?~cEZ5(6VH=7+ni!9D?`Ae7LAcE#I~C&;6mt}kEka*e>R)Ns9r^Lsz}IEP z$#K`wq z|3XiY5?iOo#P3f}d+O)X)+Gc80$-1U^K9`rBt-y@*$j95wadEZmC`!xDw3jd6U`cM zj@5yQ=~evgejy4O>OhMR7qM7uGMiH#YgqwGF@`WzmKItZh*o6A<93eH_uh87^$V$L 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candidate, Commandabilité et Observabilité} +\renewcommand{\nomentete}{UE424 - \nom} + + + +\begin{document} + +\titre{\nom} + +\cacededi{J'étais en train de chier j'ai pas vu le temps passer.}{Xavier de Tinguy de la Giroullière} + +\section*{Exercice 1: Forme quadratique de la fonction de Lyapunov} +Soit le système donné par: +\[\acc{\dot{x_1} = f_1(x_1,x_2,...,x_n)}{\dot{x_2} = f_2(x_1,x_2,...,x_n)}{\dot{x_n} = f_n(x_1,x_2,...,x_n)}\] +\begin{enumerate} +\item +L'approximation linéaire pour le point d'équilibre $x_0$ est : +\begin{align*} +\delta \dot{x}_1 =& \sum_{i=1}^{n}a_{1i}\delta x_i\\ +&\vdots\\ +\delta \dot{x}_n =& \sum_{i=1}^{n}a_{ni}\delta x_i \text{ avec, } a_{ij} = \left. \frac{\partial f_j}{\partial x_i}\right|_{x=x_0} +\intertext{Pour simplifier, on pose $x_0 = 0$, et comme $\delta x_1 = x_1 - x_{0i}$, on a:} +\dot{x}_1 =& \sum_{i=1}^{n}a_{1i} x_i\\ +&\vdots\\ +\dot{x}_n =& \sum_{i=1}^{n}a_{ni} x_i +\end{align*} + +\item \begin{align*} +\dot{z}_i =& \sum_{j=1}^n C_{ji} \dot{x_i} += \sum_{j=1}^n C_{ji} \sum_{k=1}^n a_{jk}x_k\\ +=& \sum_{k=1}^n \sum_{j=1}^n C_{ji} a_{jk}x_k += \sum_{k=1}^n \lambda_k C_{ki}x_k\\ +=& z_i \lambda_i +\end{align*} + +\item On considère la fonction de Lyapunov candidate fournie, on calcul alors: +\begin{align*} +\dot{V} =& \sum_{k=1}^n \dot{z}_k z^*_k + z_k\dot{z}^*_k\\ +=& \sum_{k=1}^n \lambda_k z_k z^*_k + \lambda_k^* z_k z^*_k\\ +=& \sum_{k=1}^n 2 Re(\lambda_k)z_kz_k^* < 0 \forall k +\end{align*} + +\item L'analyse doit se faire sur un voisinage suffisamment petit de l'origine (CN). + + +\item Modèle linéaire: +\begin{align*} +\dot{x_1} & = -x_2\\ +\dot{x_2} & = x_1\\ +\text{donc } A & = \begin{pmatrix} +0&-1\\1&0\end{pmatrix}\\ +\end{align*} +Ainsi, les valeurs propres sont i et -i.\\ +Le système linéaire est stable au sens de Lyapunov. Par contre, il n'y a pas de stabilité asymptotique.\\ + +On passe alors dans la base modale. Pour cela, on résout: +\begin{align*} +jc_{11} &= a_{11}c_{11} + a_{21}c_{21}\\ +-jc_{12} &= a_{11} c_{12} + a_{21}c_{22}\\ +jc_{21} &= a_{12} c_{11} + a_{22} c_{21} \\ +-jc_{22} &= a_{12}c_{22} + a_{22} c_{22}\\ +\Rightarrow z_1 &= x_1 + j x_2 \text{ et, } z_2 = x_2 + jx_1\\ +\end{align*} +Comme $V = \sum_{k=1}^2 z_k z_k^*$ alors, +\begin{align*} +\dot{V} &= 8(x_1^4+x_2^4)(1-\alpha x_1^2 - \beta x_2^2)\\ +\dot{V} &\leq 0\\ +\Rightarrow& \alpha x_1^2 + \beta x_2^2 > 1 +\end{align*} + +\img{0.5}{1.png} +Au voisinage de l'origine, ($\alpha x_1^2 + \beta x_2^2 >1$), c'est instable, mais en dehors de la trajectoire converge. On a un cycle limite pour $\alpha x_1^2 + \beta x_2^2 =1$ +\end{enumerate} + +\section*{Exercice II : Système du 2nd ordre} + +\begin{enumerate} +\item Suivant Routh, $a>0 \et b>0$ +\item Cas linéaire : $f(y)=by$, alors $b>0$ implique que $y(by) >0$ pour $y\neq0$, soit $yf(y)>0$ pour $y\neq 0$ +\item On prend $V(x_1,x_2) = x_2^2 + 2 \int_0^{x_1}f(\tau)d\tau$. Elle est définie positive et : \begin{align*} +\dot{V} & = 2f(x_1)\dot{x_1} + 2x_2 \dot{x_2} \\ +& = 2 x_2 f(x_1) - 2ax_2^2 -2x_2 f(x_1) +& \leq 0 \text{ : origine stable pour Lyapunov} +\end{align*} +\end{enumerate} + +\section*{Exercice III : Commandabilité et observabilité} + +\begin{enumerate} +\item On considère le système : +$ \acc{ \dot{x_1} & = -x_2 -x_2^2}{ \dot{x_2} & = u}$ + +On a donc $f(x) = \vect{ -x_2-x_2^2 \\ 0} \et g(x) = \vect{0 \\ 1}$. + +$E=\{f(x),g(x)\}$ +\begin{align*} +[f,g] & = J_gf - J_fg \\ +& = \matd{0 & 0}{0 & 0}f(x)-\matd{0 & -1 -2x_2}{0 & 0} \vect{0 \\ 1} = \vect{1 + 2x_2 \\ 0}\\ +[f,[f,g]] & = J_{[f,g]}f - J_f[f,g] \\ +& = \matd{0 & 2}{0 & 0} \vect{-x_2 -x_2^2 \\ 0} - \matd{0 & -1-2x_2}{0 & 0} \vect{ 1 + 2x_2 \\ 0} = \vect{0 \\ 0} \\ +[g,[f,g]] & = J_g[f,g] - J_{[f,g]}g \\ +& = 0 - \matd{0 & 2}{0 & 0} \vect{0 \\ 1} = -\vect{2 \\ 0} +\end{align*} +On a donc : \[ +\D = \{\vect{0 \\ 1}, \vect{2 \\ 0}, \vect{1+2x_2 \\ 0}\}\] + +$\D$ est de dimension 2 : le système est commandable. + +\item On a $f(x) = \vect{x_2^2 \\ 0} \et g(x) = \vect{0 \\ 1}$. +\begin{align*} +[f,g] & = \vect{-2 x_2 \\ 0} \\ +[f,[f,g]] & = \vect{0 \\ 0} \\ +[g,[f,g]] & = \vect{-2 \\ 0} +\end{align*} + +On a donc : \[ \D = \{\vect{0 \\ 1}, \vect{-2 \\ 0}, \vect{2x_2 \\ 0}\} \] + +$\D$ est de dimension 2 : le système est commandable.\\ + +De plus, on a $h(x) = x_1$ + +$\Vc = \{h,L_fh,L_gh,L_fL_gh,\dots \}$ et on étudie $\nabla \Vc$. + +On a $\nabla h = \vect{0 \\ 1}$. Il reste à trouver un élément de $\nabla \Vc$ qui a une 2e composante non nulle pour que $\nabla \Vc$ soit de dimension 2, et que le système soit observable. + +\begin{align*} +L_fh(x) = x_2^2 \quad & \quad \nabla L_fh(x) = \vect{0\\2x_2} \text{ ok mais dépend de $x_2$}\\ +L_gL_fh(x) = 2x_2 \quad & \quad \nabla L_gL_gh(x) = \vect{0 \\ 2} \text{ COOL!} +\end{align*} + +Le système est donc observable. +\end{enumerate} + +\end{document} \ No newline at end of file diff --git a/424-Systeme_Non_Lineaires/TD/TD5/1.png b/424-Systeme_Non_Lineaires/TD/TD5/1.png new file mode 100644 index 0000000000000000000000000000000000000000..15ac93303e65c8addeb9894483518b6a8958b851 GIT binary patch literal 54375 zcmXt9WmuG5n5Lvea-?JE5k^X+M7pI0>24*YyQI6jyOEae1`!YukS^&4>2S}tdo4de zRCMONPuz9Floh3)VUuGcAt61Jm61?ELPALgzdpr80iXZ$`jUb#sE%T?s+gFV3u{WN z;9D{$NlmBsw&qT*Mh<33A8c$tnZ0y0aWFHpar|iObd1_1goFe|l9doub;~^P)=Hq5 zb^gr$`^uZ_d#C+$>@O+VII|beN8(0h8Pyu!CxxgMsU%g$#>7^~hO{#)RV5zQeJhjs 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de temps en temps, ça calme.\\ Mais dans l'ensemble je suis un gentleman.}{Tom Colinot} + +\section*{Exercice I} + +On considère le système +\[ \accc{\dot{x_1} & = x_1 +x_2}{\dot{x_2} & = x_2^2 + u}{y & = x_1} \donc f(x) = \vect{x_1+x_2\\x_2^2}, g(x) = \vect{0 \\ 1} \et h(x) = x_1 \] + +\begin{enumerate} +\item On peut balancer $u=-x_2^2 + v$ comme des bâtards mais on va suivre le cours : + +\begin{enumerate} +\item Trouver le degré relatif +\begin{align*} +z_1 & = y = x_1 \\ +z_2 & = \dot{y} = \dot{x_1} = x_1 + x_2, \quad r>1\\ +z_3 & = \dot{z_2} = \dot{x_1} + \dot{x_2} = x_1 + x_2 + x_2^2 + u, \quad r=2 +\end{align*} + +\item +\[ \acc{\dot{z_1} = z_2}{\dot{z_2} = v} \quad \text{ modèle linéaire avec } \vect{z_1 \\ z_2} = \vect{x_1 \\ x_1 + x_2} \] + +\begin{align*} +u & = v - x_1 - x_2 - x_2^2 \\ +& = v - z_2 - ( z_2 - z_1 )^2 +\end{align*} +\end{enumerate} + +\img{0.5}{1} + +\newpage +\item \[ \acc{ \dot{z_1} & = z_2 }{ \dot{z_2} & = \ddot{y} = v = \ddot{y_r} + a_1(\dot{y_r}-\dot{y})+a_2(y_r-y)} \] + +Équation caractéristique de la dynamique +\[ x^2 + a_1 x + a_2 = 0 \] + +\img{0.5}{2} + +\item On considère maintenant le système suivant: +\[\left\{\begin{matrix} +\dot{x_1} = x_2\\ +\dot{x_2} = x_1x_2+u\\ +y = x_1 +\end{matrix} \right. \] + +Cherchons dans un premier temps uen commande linéarisante. +\begin{align*} +z_1 = y &= x_1 = h(x)\\ +z_2 = \dot{y} &= \frac{\partial h}{\partial x} \dot{x}\\ +&= \begin{pmatrix}1 &0\end{pmatrix} \begin{pmatrix} x_2 \\ x_1x_2 + u\end{pmatrix}\\ +&= x_2\\ +\ddot{y} &= \frac{\partial \dot{y}}{\partial x} \dot{x}\\ +&= \begin{pmatrix} 0 & 1\end{pmatrix} \begin{pmatrix}x_2 \\x_1x_2+u\end{pmatrix}\\ +&= x_1x_2 + u = v +\end{align*} + +Ainsi, $r=2$ et le modèle linéaire correspond à: +\[ \begin{pmatrix}z_1\\z_2\end{pmatrix} = \begin{pmatrix} +x_1 \\ x_2\end{pmatrix} \text{ et, } u = -x_1x_2 + v\] + +Pour imposer une consigne on a alors: +\begin{align*} +\ddot{\epsilon} + a_1 \dot{\epsilon} + a_0 \epsilon = 0\\ +\epsilon &= y_c - y\\ +&= y_c - z_1\\ +\dot{\epsilon} &= \dot{y_c} - z_2\\ +\ddot{\epsilon} &= \ddot{y_c} - \dot{z_2} +\intertext{Comme $\dot{z_2} = v$ alors,} +& \ddot{y_c} - \dot{z_2} + a_1 (\dot{y_c} - z_2) + a_0 ( y_c - z_1) = 0\\ + \Rightarrow& v = \ddot{y_c} + a_1(\dot{y_c} - z_2) + a_0 ( y_c - z_1) +\end{align*} +\end{enumerate} + +\section*{Exercice 2:} + +On considère le système suivant: +\[\left\{\begin{matrix} +\dot{x_1} = x_1x_3\\ +\dot{x_2} = x_1+x_2u\\ +\dot{x_3} = 1 + x_3 u +\end{matrix} \right. \] + +Examinons la commandabilité de ce système. Pour cela, on rappelle qu'il faut l'écrire sous la forme : +\[\dot{x} = f(x) + g(x) u\] +On a donc m=2 et, +\begin{align*} +&f(x) = \begin{pmatrix} x_1x_2 \\ x_1 \\1\end{pmatrix} &J_f = \begin{pmatrix}x_3 & 0 & x_1 \\ 1& 0 & 0 \\ 0 & 0 & 0\end{pmatrix}\\ +&g(x) = \begin{pmatrix} 0\\ x_2 \\x_3 \end{pmatrix} &J_g = \begin{pmatrix} +0 &0&0 \\ 0&1&0 \\ 0& 0&1\end{pmatrix} +\intertext{On calcul ensuite :} +ad_fg &= [f,g] = \begin{pmatrix}-x_1x_3 \\x_1 \\ 1\end{pmatrix}\\ +\text{donc, } J_{ad_fg} &= \begin{pmatrix}-x_3 & 0 & -x_1 \\ 1 & 0& 0 \\0 & 0& 0\end{pmatrix}\\ +\text{reste à calculer, } ad_f^2g &= [f, ad_fg] = J_{ad_f g} - J_f ad_fg = \begin{pmatrix} +-2 x_1 \\ 2x_1 x_3\\ 0 +\end{pmatrix} +\end{align*} + +Ainsi, on a $E = \{g, ad_fg, ad_f^2g\}$. Or pour $x=0$ le dernier vecteur est nul donc la dimension de E est inférieur à 3. Cela implique que le système n'est pas commandable. + + +\section*{Exercice 3:} +On considère ici le système suivant: +\[\left\{\begin{matrix} +\dot{x_1} = x_2 + u\\ +\dot{x_2} = x_1^2 + x_2\\ +\dot{x_3} = x_3 + u \\ +y = x_1 +\end{matrix} \right. \] +Déterminons la dynamique est zéros, c'est à dire que l'on va choisir $u$ de sorte à maintenir la sortie à zéro ainsi que ses dérivées successives. +Ainsi, on impose $y=0$ : +\begin{align*} +y = 0 &\Rightarrow x_1 = 0\\ +&\Rightarrow \dot{x_1} = 0\\ +&\Rightarrow x_2 + u = 0\\ +&\Rightarrow u = -x_2\\ +\text{on a aussi avec $x_1 = 0$, } &\dot{x_2} = x_2\\ +\text{et aussi, } \dot{x_3} = x_3 - x_2 +\end{align*} +On a donc la dynamique du système donnée par les valeurs propres de $ A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix}$. Les valeurs propres étant $\pm 1$, la dynamique des zéros est instable (CF début du cours de 424). + + +\end{document} \ No newline at end of file diff --git a/424-Systeme_Non_Lineaires/TD/TD6/1.png b/424-Systeme_Non_Lineaires/TD/TD6/1.png new file mode 100644 index 0000000000000000000000000000000000000000..ace595332c4e46605978aa79e23cf848bde7fa2b GIT binary patch literal 55593 zcmXtAcOaE**q7C@XOfH)C!Hh-N%jueA$w+T$<7|xqmUi4vscL8BxIAF?3ES5cX_|> z^GEMGFVFql_kCTzbqQ9Ke@cW;gO7oMK_o3Du8e_!MGt>Ji@=0GkFaBw;4fIOM5R>_ z2*lL9!W{hL4Mz!e$LF>tj?M-S#u%nHw${e?Ul}SCH zZ@d|MM?2>FX1V)G|CS*W`E?8tUwvF^?2ruQ?y8r2B5V_?o+Wx4EOkA_=M)=Pg(k9C z@?^@ypP$@SqyPU6c!I3BlTgy+9e&y79bDi1@hC`0z6E!Q-!bD(3YYXVp3uE%sLdZG7DkEPY*?yD zIb!)t_5-^?3w)4y%Uv;6CTq}af0B^mNPtVuk~^uss%2n+OKzw_SL?seNH+V`+9^8K z)sJz1a3LKq`KJ(rRE8^#nQKEch^k87h zgZjBX#na-&L=U-Y|8#XZ@>*AFPTM%Yu;SRY6CL7`juBVdcdM_jYHuUn5mtM0`N-n7 z%{(dc1tXJ|oU%WUKAKM&mhMmDQ}Fq%zDT(I&toD>d#>d(E#g5)=@?^;q`uMrdrQLX z_h=a86}jY+B&;uA%H?MbaygtR>Tt8iaR2XZkTAR2 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zSM_=>=TP1l-B+(jfFG6eU_Zp6$5dXGih%$TAQmPbQ1C&7NT8<;R`=eL-uCK<2yu>& zcg#7hRQFc|eH(LCn5AZzi2AZuGCi%$`c?nX0ve5O7A+G$GF;y2{uP|D&=4{51o^ap zEJCCNwytYewhPmE4*mD@;l9)d`XaL%Q+Q8v5sqz`?63HCMu6Kg{RvnnAZZ4EzC@z0$(}!>V(J{x=6!a?n3L;OU)t0rC)MD9JOdeWi&c$ougMvB2yV*xONy{Tx3cs0 z?^4k#T=X_z?2yO{cPvqe{qVj{B^khh#<>iCFw%xUcKC9wQRIb-7nzut&{R|Il%50= zF5zLd{CAoBv7WLI?w0D@irxMoGqn|}F!`{gWQXp*!ENhqp2XgCG9t`6y(hVHWPqzo z({WfRh76IUdD4$8-MG1)jX>#IX3m?t$Zf$|8F;r$8QCcX!upUBS;lzv-CD83=^Fx) zoxGalJ`5@c5_;(uty8k;O6|mPJ8)1&98x-+f`p#zY)w#lGouWsz_+~8 z^t|$Z1fLNv)1GUn)89iR&;(bUpx*Gfnfw?HzGpE1?;mq4nNHgOdlm{WhPBRo#a;Km dpHLXr)UFu*N%_EMKDq%feQiUnd`;_+{{tO?);j 0$, sachant que $\epsilon = \omega_{opt} - \omega_t$, il s'agit donc de remplacer $\epsilon$ dans un premier temps, on trouve : +\[\dot{\omega}_{opt} - \dot{\omega_t} + a_0(\omega_{opt} - \omega_t)\] + +Si l'on injecte ceci dans l'équation dynamique, on trouve alors la commande qui suit la restriction imposée sur $\epsilon$ : +\begin{align*} +\dot{\omega}_{opt} - \frac{T_a}{J_t} + \frac{K_t}{J_t} \omega_t + \frac{T_g}{J_t} + a_0((\omega_{opt} - \omega_t) = 0\\ +T_g = -a_0 J_t\omega_{opt} - J_t \dot{\omega}_{opt}v + (a_0 J_t - K_t)\omega_t + T_a +\end{align*} +$T_a$ étant le terme à linéariser. + +\item Si l'on impose une perturbation constante d, la commande précédente reproduit la perturbation et ne la rejette pas. L'équation obtenue est: +\[\dot{\omega}_{opt} - \frac{T_a}{J_t} + \frac{K_t}{J_t} \omega_t + \frac{T_g}{J_t} + a_0((\omega_{opt} - \omega_t) = - \frac{d}{J_t} \] +On voit bien que la solution de cette équation aura un terme constant dépendant de d. + +\item On se propose d'imposer une convergence asymptotique vers 0 suivant la dynamique $\ddot{\epsilon} + a_1\dot{\epsilon} + a_0 \epsilon = 0$ avec $a_1 > 0 $ et $a_0 >0$. Comme précédemment, $\epsilon = \omega_{opt} - \omega_t$. Donc en injectant ceci on a: +\[(\ddot{\omega}_{opt} - \ddot{\omega_t}) + a_1 (\dot{\omega}_{opt} - \dot{\omega}_t) + a_0 (\omega_{opt} - \omega_t) = 0\] + +On va avoir besoin de dériver l'équation dynamique pour remplacer $\ddot{\omega_t}$,et on trouve : +\[\dot{T_g} = \dot{T_a} - K_t\dot{\omega_t} - J_t a_1(\dot{\omega}_{opt} - \dot{\omega_t}) - J_ta_0(\omega_{opt} - \omega_t) - J_t \ddot{\omega}_{opt}\] + +On constate que l'on doit introduire un capteur pour mesurer $\dot{\omega_t}$. + +On a donc un modèle d'asservissement suivant le schéma suivant: +%\img{0.5}{1} +Pour une perturbation constante d, on a bien toujours la dynamique sur $\epsilon$. d disparaissant lors du calcul de la dérivée de $\dot{\omega_t}$. + +\item La partie que l'on souhaite linéariser dans la commande de $T_g$ (respectivement $\dot{T_g}$ pour la question 3) est celle contenant les termes dépendant de $\omega_{opt}$. Il suffit donc d'égaler ces termes à une commande $v$ puis d'exprimer cette commande en fonction de $T_g$ comme vu dans les TDs précédents. + +\end{enumerate} +\end{document} \ No newline at end of file diff --git a/424-Systeme_Non_Lineaires/TD/TD8/TD8.tex b/424-Systeme_Non_Lineaires/TD/TD8/TD8.tex new file mode 100644 index 0000000..5e6731e --- /dev/null +++ b/424-Systeme_Non_Lineaires/TD/TD8/TD8.tex @@ -0,0 +1,116 @@ +\documentclass{article} +\input{../../preambule/preambule} + +\newcommand{\nom}{TD8 : Commande hiérarchisée et Robustesse} +\renewcommand{\nomentete}{UE424 - \nom} + + +\begin{document} + + +\titre{\nom} +\section*{Exercice I: Platitude} +\begin{enumerate} +\item On considère le système suivant: +\[\acc{\dot{x_1} &= x_2 - x_1 cos x_1}{\dot{x_2} &= (x_2^2 + u)(5 + sinx_1)} \] + +Montrons que $x_1$ est une sortie plate, pour cela, il faut exprimer u en fonction de $x_1$ et ses dérivées uniquement: +\begin{align*} +x_2 &= \dot{x_1} + x_1 cos x_1\\ +u &= \frac{\dot{x_2}}{5 + sin x_1} - x_2^2\\ +&= \frac{\ddot{x_1} + \dot{x_1}cosx_1 - \dot{x_1}x_1 sinx_1}{5 + sin x_1} - (\dot{x_1} + x_1 cos x_1)^2 +\end{align*} +$x_1$ est bien une sortie plate. + +\item On considère le système suivant: +\[ \left\{\begin{matrix} +\dot{x_1} &= -x_1^2 + x_2\\ +\dot{x_2} &= x_2x_1 + u_1\\ +\dot{x_3} &= u_2 +\end{matrix}\right. \] +Montrons que $x_1$ et $x_3$ sont des sorties plates: +\begin{align*} +x_2 &= \dot{x_1}- x_1^2 \\ +u_1 &= \dot{x_2} - x_2 x_1\\ +&= \ddot{x_1} - 3x_1 \dot{x_1} + x_1^3\\ +u_2 &= \dot{x_3} +\end{align*} +On a bien les commandes en fonctions de $x_1$,$x_3$ et leurs dérivées uniquement. + +\end{enumerate} + +\section*{Exercice II: Planification} +On considère le système suivant: +\[\acc{\dot{x_1} &= x_2}{\dot{x_2} &= \alpha\dot{x_1} + u} \] +\begin{enumerate} +\item Trouvons la sortie plate y. On remarque que pour $y=x_1$ on a $u = \ddot{y} - \alpha \dot{y}$, donc ce y convient (est une sortie plate) et on a alors le système: +\[ \left\{\begin{matrix} +\dot{x_1} &= y\\ +\dot{x_2} &= \dot{y}\\ +u &= \ddot{y} - \alpha \dot{y} +\end{matrix}\right. \] + +\item +\begin{align*} +y_c(t) &= a(T-t)^3 + b(T-t)^2 + c(T-t) + d\\ +y_c(0) &= y_0 \Rightarrow aT^3 + bT^2 + cT + d = y_0\\ +y_c(T) &= y_T \Rightarrow d = y_T\\ +\dot{y_c}(T) &= 0 \Rightarrow -c = 0\\ +\ddot{y_c}(T) &= 0 \Rightarrow b = 0\\ +&\Rightarrow aT^3 = y_0 - y_T\\ +&\Rightarrow a = \frac{y_0 - y_T}{T^3}\\ +u &= \ddot{y_c} - \alpha \dot{y_c}\\7 +&=3\frac{y_0 -y_T}{T^3}(T-t)(3 + \alpha(T-t) +\end{align*} +On a la commande en BO. En posant $ \delta x = x - x_c$ on peut linéariser le modèle autour du point et approcher une trajectoire point par point.\\ +La planification de la trajectoire permet de trouver un modèle linéaire autour de la trajectoire obtenue via $u_c$, la commande de planification. + +\end{enumerate} + +\section*{Exercice III: Suspension magnétique} +On peut appliquer le backstepping car le système est de forme triangulaire : +\[ \left\{ \begin{matrix} +\dot{x_1} &= x_2\\ +\dot{x_2} &= -g + \frac{k}{m} \frac{x_3^2}{(c-x_1)^2}\\ +\dot{x_3} &= \frac{-x_3 + k_v u}{\tau} +\end{matrix} \right. \] + +On a $u^*$ qui permet d'avoir $x_3$ qui via $\alpha_3$ donne $x_3^*$ qui lui donne $x_2$ qui via $\alpha_2$ donne $x_2^*$ qui donne $x_1$ qui lui donne $x_1^*$ via $\alpha_1$. +Peut-être. + +\subsection*{Etape 1:} +On pose $V_1(x_1) = \frac{1}{2}(x_1-x_1^*)^2$ avec $x_1^* = z_*$, et, $\dot{V_1} = \alpha_1 (x_1 - x_1^*)^2$. En égalisant les deux termes, on trouve après simplification que: +\[ x_2^* = \alpha_1(x_1 - x_1^*) + \dot{x_1}^* \text{avec, } \alpha_1 <0\] + +\subsection*{Etape 2:} +On pose: +\begin{align*} +V_2(x_1,x_2)& = \frac{1}{2}(x_1 - x_1^*)^2 + \frac{1}{2}(x_2 - x_2^*)^2\\ +\dot{V_2}(x_1,x_2) &= \alpha_1(x_1 - x_1^*)^2 + \alpha_2(x_2 - x_2^*)^2 \text{avec, } \alpha_2 < \alpha_1 <0 +\intertext{On dérive, on égalise et on injecte $\dot{x_2}$ pour trouver} +x_3^2 &= \hat{x_3}^* = (\alpha_2(x_2-x_2^*) + g + \dot{x_2}^*) \frac{m}{k} (c-x_1)^2 +\end{align*} + +\subsection*{Etape 3:} +On s'intéresse ici à $\hat{x_3}$ plutot que $x_3^2$. +On pose: +\begin{align*} +V_3(x_1,x_2,\hat{x_3}) &= V_2(x_1,x_2) + \frac{1}{2}(\hat{x_3} - \hat{x_3}^*)^2\\ +\dot{V_3}(x_1,x_2,\hat{x_3}) &= \dot{V_2}(x_1,x_2) + \alpha_3(\hat{x_3} - \hat{x_3}^*)^2 +\intertext{On dérive, on égalise et on injecte $\dot{x_3}$ pour trouver} +u = \frac{2x_3 + \alpha_3 \tau (x_3 - \hat{x_3}^*/x_3) + \tau \dot{\hat{x_3}}^*}{2k_v} +\end{align*} + +Il faut éviter que $x_3 = 0$ pour avoir i non nul? + +\section*{Exercice IV: Commande par modes glissants} +$\alpha_0 = 1.5, \alpha(t) = \alpha_0 + \Delta \alpha avec |\Delta\alpha| \leq 0.5$ +$S = \dot{\epsilon} + \beta_0 \epsilon$ + +$\dot{S} = \ddot{\epsilon} + \beta_0 \dot{\epsilon}$ Poursuite assymptotique +$u = (\alpha_0 x_2^2 + \ddot{y_c} + \dot{S} + \alpha K sign(S))$ +Premier terme: mode linéarisant +Terme 3 et 4: mode glissant + +$\alpha >1 K tel que |\Delta \alpha x_2^2| < K$ +\end{document} \ No newline at end of file