From 017b3267e7125c7716100964fb79993102ed53e8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Fri, 16 Feb 2024 19:00:20 +0100 Subject: [PATCH] added new control theory --- .github/workflows/buildpdf.yml | 10 +-- .../Introduction_to_control_theory.tex | 67 ++++++++++++++++++- preamble_general.sty | 2 +- 3 files changed, 72 insertions(+), 7 deletions(-) diff --git a/.github/workflows/buildpdf.yml b/.github/workflows/buildpdf.yml index 047b56f..a02a1b8 100644 --- a/.github/workflows/buildpdf.yml +++ b/.github/workflows/buildpdf.yml @@ -182,11 +182,11 @@ jobs: with: root_file: Continuous_optimization.tex working_directory: Mathematics/5th/Continuous_optimization/ - # - name: Compile - IEPDE - # uses: xu-cheng/latex-action@v2 - # with: - # root_file: Introduction_to_evolution_PDEs.tex - # working_directory: Mathematics/5th/Introduction_to_evolution_PDEs/ + - name: Compile - IEPDE + uses: xu-cheng/latex-action@v2 + with: + root_file: Introduction_to_control_theory.tex + working_directory: Mathematics/5th/Introduction_to_control_theory/ - name: Compile - INEPDE uses: xu-cheng/latex-action@v2 with: diff --git a/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex b/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex index c200c11..234f7b8 100644 --- a/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex +++ b/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex @@ -94,7 +94,6 @@ \begin{theorem} If the origin is asymptotically stable, then its basin of attraction is an open set included in $\mathcal{O}$. Besides, $\exists \beta_\mathcal{A}\in \mathcal{KL}$ such that $\forall \vf{x}_0\in\mathcal{A}$, any solution $\vf{X}(\vf{x}_0, \cdot)$ exists for all $t\geq 0$ and satisfies $\omega_{\mathcal{A}}(\norm{\vf{X}(\vf{x}_0, t)})\leq \beta_\mathcal{A}(\norm{\vf{x}_0}, t)$ for all $t\geq 0$. \end{theorem} - \subsubsection{Sufficient conditions for stability} \begin{theorem} Assume that $\vf{f}\in\mathcal{C}^1$. Then: \begin{enumerate} @@ -108,6 +107,72 @@ \begin{corollary} If $\vf{f}\in\mathcal{C}^1$ and $\vf{Df}(\vf{0})$ has all its eigenvalues with negative real part, then the origin is asymptotically stable. \end{corollary} + \begin{theorem} + Let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function which is positive definite on $\mathcal{O}$. Then, if + $$ + D_f^+V(\vf{x}) = \limsup_{t\to 0^+}\frac{V(\vf{x} + \vf{f}(\vf{x})t) - V(\vf{x})}{t} + $$ + is non-positive for all $\vf{x}\in \mathcal{O}$, then the origin is stable. The function $V$ is called a \emph{Lyapunov function}. + \end{theorem} + \begin{proof} + Since $\mathcal{O}$ is a neighbourhood of the origin $\exists R>0$ such that $\overline{B(0,R)}\subseteq \mathcal{O}$. Then, since $V$ is continuous and positive definite, $\exists \alpha_1,\alpha_2\in\mathcal{K}$ such that $\alpha_1(\norm{\vf{x}})\leq V(\vf{x})\leq \alpha_2(\norm{\vf{x}})$ for all $\vf{x}\in B(0,R)$. Let $\mu:={\alpha_2}^{-1}(\alpha_1(R/2))$. Then, any solution with initial conditions $\norm{\vf{x}_0}<\mu$ belongs to $\overline{B(0,R)}$ at least for $t\in[0, T)$. Now if we consider $v(t) := V(\vf{X}(\vf{x}_0, t))$, then we have $\dot{v}(t) = D_f^+V(\vf{X}(\vf{x}_0, t))\leq 0$ for all $t\geq 0$. Thus, $\forall t \in [0, T)$ we have: + \begin{multline*} + \alpha_1(\norm{\vf{X}(\vf{x}_0, t)})\leq V(\vf{X}(\vf{x}_0, t))=v(t) \leq v(0)=\\ = V(\vf{x}_0) \leq \alpha_2(\norm{\vf{x}_0}) + \end{multline*} + And so $\norm{\vf{X}(\vf{x}_0, t)}\leq \alpha_2^{-1}(\alpha_1(\norm{\vf{x}_0}))\leq R/2$ for all $t\in[0, T)$. This mean that in fact $T=\infty$ and so the origin is stable. + \end{proof} + \begin{theorem} + Let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function which is positive definite on $\mathcal{O}$. Then, if + $$ + D_f^+V(\vf{x})\leq -w(\vf{x}), \quad \forall \vf{x}\in \mathcal{O} + $$ + with $w:\mathcal{O}\to \RR_{\geq 0}$ continuous and positive definite, then the origin is globally asymptotically stable. + \end{theorem} + \begin{theorem}[Lasaalle's invariance principle] + Let $K$ be a compact set contained in $\mathcal{O}$ and let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function which is positive definite on $\mathcal{O}$ and such that $D_f^+V(\vf{x})\leq -w(\vf{x})$ for all $\vf{x}\in K$ with $w:\mathcal{O}\to \RR_{\geq 0}$ continuous (not necessarily positive definite). Then, for any solution $\vf{X}(\vf{x}_0, \cdot)$ with $\vf{x}_0\in K$ and defined on $K$ for all $t\geq 0$, $\exists v^*\in\RR_{\geq 0}$ such that $\vf{X}(\vf{x}_0, t)$ converges to the largest positively invariant set contained in: + $$ + \{\vf{y}\in K: V(\vf{y})=v^*\text{ and }w(\vf{y})=0\} + $$ + \end{theorem} + \begin{theorem}[Chetaev's theorem] + Let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function such that: + \begin{itemize} + \item $0\in \Fr{G}$, with $G:=\{\vf{x}\in \mathcal{O}: V(\vf{x})=0\}$. + \item There exists a neighbourhood $U$ (called Chetaev surface) of the origin such that $D_f^+V(\vf{x})>0$ for all $\vf{x}\in U\cap G$ + \end{itemize} + Then, the origin is unstable. + \end{theorem} + \begin{theorem} + If the origin is asymptotically stable, then $\forall\varepsilon>0$ $\{f(\vf{x}): \norm{\vf{x}}\leq \varepsilon\}$ is a neighbourhood of the origin. + \end{theorem} + \begin{theorem} + If the origin is locally asymptotically stable with basin of attraction $\mathcal{A}$, then $\exists \lambda>0$ and $v:\mathcal{A}\to \RR_{\geq 0}$ $\mathcal{C}^\infty$, positive definite and proper (that is, $\displaystyle \lim_{d(\vf{x}, \Fr{\mathcal{A}})\to 0}v(\vf{x})=\infty$) such that: + $$ + D_f^+v(\vf{x})\leq -\lambda v(\vf{x})\quad\forall \vf{x}\in \mathcal{A} + $$ + \end{theorem} + \subsubsection{Control design and stabilization of equilibrium points} + \begin{definition} + The system $\dot{\vf{x}} = \vf{f}(\vf{x}, \vf{u})$ is said to be \emph{controllable} in time $T>0$ if $\forall \vf{x}_0, \vf{x}_T\in \mathcal{O}$ $\exists \vf{u}: [0, T]\to \RR^p$ such that the solution $\vf{X}(\vf{x}_0,\cdot, \vf{u})$ of the system with initial condition $\vf{X}(\vf{x}_0, 0,\vf{u}) = \vf{x}_0$ satisfies $\vf{X}(\vf{x}_0, T,\vf{u}) = \vf{x}_T$. + \end{definition} + \begin{definition} + The origin is said to be \emph{asymptotically stabilizable} if there exists $q\in \NN$, a neighbourhood $\mathcal{V}\subseteq \RR^q$ of the origin and $\vf\varphi:\RR\times\RR^n\times\mathcal{V}\to \RR^q$, $\vf\psi:\RR\times\RR^n\times\mathcal{V}\to \RR^p$ both continuous, such that the origin is an asymptotically stable solution of the system: + $$ + \begin{cases} + \dot{\vf{x}} = \vf{f}(\vf{x}, \vf{u}) \\ + \dot{\vf{u}} = \vf\varphi(t, \vf{x}, \vf\chi) \\ + \vf{\vf\chi} = \vf\psi(t, \vf{x}, \vf\chi) + \end{cases} + $$ + \end{definition} + \begin{theorem}[Kalmann's theorem] + Consider the linear system $\dot{\vf{x}} = \vf{Ax} + \vf{Bu}$ with $\vf{A}\in \RR^{n\times n}$ and $\vf{B}\in \RR^{n\times p}$. Then, the origin is asymptotically stabilizable if and only if + $$ + \rank\vf{C}:=\rank\begin{pmatrix} \vf{B} & \vf{AB} & \cdots & \vf{A}^{n-1}\vf{B} \end{pmatrix} = n + $$ + The matrix $\vf{C}$ is called the \emph{controllability matrix}. + \end{theorem} + \subsection{Control theory in PDEs} \end{multicols} \end{document} \ No newline at end of file diff --git a/preamble_general.sty b/preamble_general.sty index 01c9fb9..40f01f6 100644 --- a/preamble_general.sty +++ b/preamble_general.sty @@ -74,7 +74,7 @@ {NMPDE}{\apl} % Numerical methods for pdes {SCO}{\pro} % Stochastic control {JP}{\pro} % Jump processes - {ICT}{\ana} % Introduction to control theory + {ICT}{\apl} % Introduction to control theory }{\col}% } \ExplSyntaxOff