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+\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}
generated by cgit v1.2.1 (git 2.18.0) at 2024-06-29 11:18:56 +0200