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author | Joshua Baer <joshua.baer@ost.ch> | 2022-08-16 10:28:39 +0200 |
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committer | Joshua Baer <joshua.baer@ost.ch> | 2022-08-16 10:28:39 +0200 |
commit | f3fc1867221ab08c20cbb3f21663e89a10526dcb (patch) | |
tree | 49ca3a3a863f57f2ea69025ed493ba6f35d12c36 /buch/papers/kra/presentation/presentation.tex | |
parent | signal m draw (diff) | |
parent | fix a typo (diff) | |
download | SeminarSpezielleFunktionen-f3fc1867221ab08c20cbb3f21663e89a10526dcb.tar.gz SeminarSpezielleFunktionen-f3fc1867221ab08c20cbb3f21663e89a10526dcb.zip |
Merge branch 'master' of github.com:JODBaer/SeminarSpezielleFunktionen
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-rw-r--r-- | buch/papers/kra/presentation/presentation.tex | 491 |
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diff --git a/buch/papers/kra/presentation/presentation.tex b/buch/papers/kra/presentation/presentation.tex new file mode 100644 index 0000000..eb6541b --- /dev/null +++ b/buch/papers/kra/presentation/presentation.tex @@ -0,0 +1,491 @@ +\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer} + +% style +\mode<presentation>{ + \usetheme{Frankfurt} +} +%packages +\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-} +\usepackage[english]{babel} +\usepackage{graphicx} +\usepackage{array} + +\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} +\usepackage{ragged2e} + +\usepackage{bm} % bold math +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{multirow} % multi row in tables +\usepackage{booktabs} %toprule midrule bottomrue in tables +\usepackage{scrextend} +\usepackage{textgreek} +\usepackage[rgb]{xcolor} + +\usepackage[normalem]{ulem} % \sout + +\usepackage{ marvosym } % \Lightning + +\usepackage{multimedia} % embedded videos + +\usepackage{tikz} +\usepackage{pgf} +\usepackage{pgfplots} + +\usepackage{algorithmic} + +%citations +\usepackage[style=verbose,backend=biber]{biblatex} +\addbibresource{references.bib} + + +%math font +\usefonttheme[onlymath]{serif} + +%Beamer Template modifications +%\definecolor{mainColor}{HTML}{0065A3} % HSR blue +\definecolor{mainColor}{HTML}{D72864} % OST pink +\definecolor{invColor}{HTML}{28d79b} % OST pink +\definecolor{dgreen}{HTML}{38ad36} % Dark green + +%\definecolor{mainColor}{HTML}{000000} % HSR blue +\setbeamercolor{palette primary}{bg=white,fg=mainColor} +\setbeamercolor{palette secondary}{bg=orange,fg=mainColor} +\setbeamercolor{palette tertiary}{bg=yellow,fg=red} +\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic +\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points) +\setbeamercolor{section in toc}{fg=black} % TOC sections +\setbeamertemplate{section in toc}[sections numbered] +\setbeamertemplate{subsection in toc}{% + \hspace{1.2em}{$\bullet$}~\inserttocsubsection\par} + +\setbeamertemplate{itemize items}[circle] +\setbeamertemplate{description item}[circle] +\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true] +\beamertemplatenavigationsymbolsempty + +\setbeamercolor{footline}{fg=gray} +\setbeamertemplate{footline}{% + \hfill\usebeamertemplate***{navigation symbols} + \hspace{0.5cm} + \insertframenumber{}\hspace{0.2cm}\vspace{0.2cm} +} + +\usepackage{caption} +\captionsetup{labelformat=empty} + +%Title Page +\title{KRA} +\subtitle{Kalman Riccati Abel} +\author{Samuel Niederer} +% \institute{OST Ostschweizer Fachhochschule} +% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}} +\date{\today} + +\input{../packages.tex} + +\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}% + +\newcommand*{\HL}{\textcolor{mainColor}} +\newcommand*{\RD}{\textcolor{red}} +\newcommand*{\BL}{\textcolor{blue}} +\newcommand*{\GN}{\textcolor{dgreen}} +\newcommand{\dt}[0]{\frac{d}{dt}} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + + +\makeatletter +\newcount\my@repeat@count +\newcommand{\myrepeat}[2]{% + \begingroup + \my@repeat@count=\z@ + \@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}% + \endgroup +} +\makeatother + +\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings} + +\begin{document} + +\begin{frame} + \titlepage +\end{frame} + +\begin{frame} + \frametitle{Content} + \tableofcontents +\end{frame} + +\section{Einführung} + +\begin{frame} + \begin{itemize} + \item<1|only@1> \textbf{K}alman + \item<1|only@1> \textbf{R}iccati + \item<1|only@1> \textbf{A}bel + + \item<2|only@2> \textcolor{red}{\sout{\textbf{K}alman}} + \item<2|only@2> \textbf{R}iccati + \item<2|only@2> \textbf{A}bel + + \item<3|only@3> \textcolor{red}{\sout{\textbf{K}alman}} \textcolor{green}{Federmassesytem} + \item<3|only@3> \textbf{R}iccati + \item<3|only@3> \textbf{A}bel + + \item<4|only@4> \textcolor{red}{\sout{\textbf{K}alman}} \textcolor{green}{Federmassesytem} + \item<4|only@4> \textbf{R}iccati + \item<4|only@4> \uwave{\textbf{A}bel} + \end{itemize} +\end{frame} + +\section{Riccati} + +\begin{frame} + \frametitle{Riccatische Differentialgleichung} + \begin{equation*} + % y'(x) = f(x)y^2(x) + g(x)y(x) + h(x) + x'(t) = f(t)x^2(t) + g(t)x(t) + h(t) + \end{equation*} + + \pause + + \begin{equation*} + \dot{X}(t) = - X(t)BX(t) - X(t)A + DX(t) + C + \end{equation*} + + % \pause + % Anwendungen + % \begin{itemize} + % \item Zeitkontinuierlicher Kalmanfilter + % \item Regelungstechnik LQ-Regler + % \item Federmassesyteme + % \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{Auftreten der Gleichung} + \begin{columns} + \column{0.4 \textwidth} + \begin{equation*} + \dt + \begin{pmatrix} + X \\ + Y + \end{pmatrix} + = + \underbrace{ + \begin{pmatrix} + A & B \\ + C & D + \end{pmatrix} + }_{H} + \begin{pmatrix} + X \\ + Y + \end{pmatrix} + \end{equation*} + + \pause + + \column{0.4 \textwidth} + \begin{equation*} + U = YX^{-1} \qquad \dt U = ? + \end{equation*} + \end{columns} + + \pause + + \begin{align*} + \dt U & = \dot{Y} X^{-1} + Y \dt X^{-1} \\ + \uncover<4->{ & = (CX + DY) X^{-1} - Y (X^{-1} \dot{X} X^{-1})\\} + \uncover<5->{ & = C\underbrace{XX^{-1}}_\text{I} + D\underbrace{YX^{-1}}_\text{U} - Y(X^{-1} (AX + BY) X^{-1})\\} + \uncover<6->{ & = C + DU - \underbrace{YX^{-1}}_\text{U}(A\underbrace{XX^{-1}}_\text{I} + B\underbrace{YX^{-1}}_\text{U})\\} + \uncover<7->{ & = C + DU - UA - UBU} + \end{align*} +\end{frame} + +\begin{frame} + \frametitle{Lösen der Gleichung} + \begin{equation*} + \begin{pmatrix} + X(t) \\ + Y(t) + \end{pmatrix} + = + \Phi(t_0, t) + \begin{pmatrix} + I(t) \\ + U_0(t) + \end{pmatrix} + = + \begin{pmatrix} + \Phi_{11}(t_0, t) & \Phi_{12}(t_0, t) \\ + \Phi_{21}(t_0, t) & \Phi_{22}(t_0, t) + \end{pmatrix} + \begin{pmatrix} + I(t) \\ + U_0(t) + \end{pmatrix} + \end{equation*} + + \pause + + \begin{equation*} + U(t) = + \begin{pmatrix} + \Phi_{21}(t_0, t) + \Phi_{22}(t_0, t) U_0(t) + \end{pmatrix} + \begin{pmatrix} + \Phi_{11}(t_0, t) + \Phi_{12}(t_0, t) U_0(t) + \end{pmatrix} + ^{-1} + \end{equation*} + + \pause + + % wobei $\Phi(t, t_0)$ die sogennante Zustandsübergangsmatrix ist. + + \begin{equation*} + \Phi(t_0, t) = e^{H(t - t_0)} + \end{equation*} +\end{frame} + +\section{Federmassystem} +\begin{frame} + \frametitle{Federmassesystem} + \begin{columns} + \column{0.5 \textwidth} + \input{../images/simple_mass_spring.tex} + + \column{0.5 \textwidth} + \begin{align*} + \uncover<2->{F_R & = k \Delta_x \\} + \uncover<3->{F_a & = am = \ddot{x} m \\} + \uncover<4->{F_R & = F_a \Leftrightarrow k \Delta_x = \ddot{x} m\\} + \uncover<5->{\ddot{x} & = \frac{k \Delta_x}{m} \\} + \uncover<6->{x(t) & = A \cos(\omega_0 + \Phi), \quad \omega_0 = \sqrt{\frac{k}{m}}} + \end{align*} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Phasenraum} + \begin{columns} + \column{0.3 \textwidth} + \begin{tikzpicture}[scale=3] + \draw[->, thick] (0, 0) -- (1,0) node[right] {$q$}; + \draw[->, thick] (0.5, -0.5) -- (0.5,0.5) node[above]{$p$}; + \end{tikzpicture} + \column{0.7 \textwidth} + Impulskoordinaten $p = (p_{1}, p_{2}, ..., p_{n}), \quad p=mv$ \\ + Ortskoordinaten $q = (q_{1}, q_{2}, ..., q_{n})$ \\ + + + + \begin{align*} + \uncover<2->{\mathcal{H}(q, p) & = \underbrace{T(p)}_{E_{kin}} + \underbrace{V(q)}_{E_{pot}} = E_{tot} \\} + \uncover<3->{ & = \frac{p^2}{2m}+ \frac{k q^2}{2}} + \end{align*} + + + + \begin{equation*} + \uncover<4->{ + \dot{q_{k}} = \frac{\partial \mathcal{H}}{\partial p_k} + \qquad + \dot{p_{k}} = -\frac{\partial \mathcal{H}}{\partial q_k} + } + \end{equation*} + + \pause + + \begin{equation*} + \uncover<5->{ + \begin{pmatrix} + \dot{q} \\ + \dot{p} + \end{pmatrix} + = + \begin{pmatrix} + 0 & \frac{1}{m} \\ + -k & 0 + \end{pmatrix} + \begin{pmatrix} + q \\ + p + \end{pmatrix} + } + \end{equation*} + + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Phasenraum} + \input{../images/phase_space.tex} +\end{frame} + +\begin{frame} + \frametitle{Federmassesystem} + \begin{columns} + \column{0.6 \textwidth} + \scalebox{0.8}{\input{../images/multi_mass_spring.tex}} + \begin{align*} + \uncover<2->{\mathcal{H} & = T + V \\} + \uncover<7->{ & = \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2} + \frac{k_1 q_1^2}{2} + \frac{k_c (q_2 - q_1)^2}{2} + \frac{k_2 q_2^2}{2}} + \end{align*} + + \column{0.4 \textwidth} + \begin{align*} + \uncover<3->{T & = T_1 + T_2} \\ + \uncover<5->{ & = \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2} } \\ + \uncover<4->{V & = V_1 + V_c + V_2 } \\ + \uncover<6->{ & = \frac{k_1 q_1^2}{2} + \frac{k_c (q_2 - q_1)^2}{2} + \frac{k_2 q_2^2}{2}} + \end{align*} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Federmassesystem} + \begin{equation*} + \begin{pmatrix} + \dot{q_1} \\ + \dot{q_2} \\ + \dot{p_1} \\ + \dot{p_2} \\ + \end{pmatrix} + = + \begin{pmatrix} + 0 & 0 & \frac{1}{2m_1} & 0 \\ + 0 & 0 & 0 & \frac{1}{2m_2} \\ + -(k_1 + k_c) & k_c & 0 & 0 \\ + k_c & -(k_c + k_2) & 0 & 0 \\ + \end{pmatrix} + \begin{pmatrix} + q_1 \\ + q_2 \\ + p_1 \\ + p_2 \\ + \end{pmatrix} + \Leftrightarrow + \dt + \begin{pmatrix} + Q \\ + P \\ + \end{pmatrix} + \underbrace{ + \begin{pmatrix} + 0 & M \\ + K & 0 + \end{pmatrix} + }_{H} + \begin{pmatrix} + Q \\ + P \\ + \end{pmatrix} + \end{equation*} + + \pause + + $U = PQ^{-1} \qquad \dt U = ?$ + + \pause + + \begin{align*} + \dt U & = C + DU - UA - UBU \\ + & = K - UMU + \end{align*} + +\end{frame} + +\begin{frame} + \frametitle{Einfluss der Anfangsbedingung:} + \begin{columns} + \column{0.4 \textwidth} + \begin{equation*} + \uncover<2->{q_0 = + \begin{pmatrix} + q_{10} \\ + q_{20} + \end{pmatrix} + = + \begin{pmatrix} + 3 \\ + 1 + \end{pmatrix} + } + \end{equation*} + \begin{equation*} + \uncover<3->{q_0 = + \begin{pmatrix} + q_{10} \\ + q_{20} + \end{pmatrix} + = + \begin{pmatrix} + 3 \\ + 3 + \end{pmatrix} + } + \end{equation*} + \begin{equation*} + \uncover<4->{q_0 = + \begin{pmatrix} + q_{10} \\ + q_{20} + \end{pmatrix} + = + \begin{pmatrix} + 2 \\ + -2 + \end{pmatrix} + } + \end{equation*} + \column{0.6 \textwidth} + \scalebox{0.8}{\input{../images/multi_mass_spring.tex}} + \end{columns} +\end{frame} + +\section{Schlussteil} +\begin{frame} + \frametitle{Zusammenfassung} + \begin{itemize} + \pause + \item{Riccatische Differentialgleichung} + \pause + \begin{itemize} + \item{Ausgansgleichung} + \pause + \item{Lösung} + \end{itemize} + \pause + \item{Harmonischer Ozillator} + \pause + \begin{itemize} + \item{Hamiltonfunktion} + \pause + \item{Phasenraum} + \end{itemize} + \pause + \item{Gekoppelter harmonischer Ozillator} + \pause + \begin{itemize} + \item{Riccatische Differentialgleichung} + \pause + \item{Einfluss der Anfangsbedingungen} + \end{itemize} + \pause + \item{\uwave{Abel}} + \begin{itemize} + \pause + \item{Nichtlineare Federkonstante} + \end{itemize} + + \end{itemize} +\end{frame} + +\end{document} |