From f77860148136e93d0cce26c85232b7900f41e355 Mon Sep 17 00:00:00 2001 From: Leif Andersson <leif.andersson@control.lth.se> Date: Mon, 2 May 2022 11:34:25 +0200 Subject: [PATCH] Initial files --- .gitignore | 307 + README.md | 89 +- beamerExample.tex | 603 + figures/count_cont.eps | 4648 ++ figures/count_cont.png | Bin 0 -> 98080 bytes figures/guck.eps | 3365 ++ figures/lyap.tex | 21 + figures/lyap_ex.tex | 19 + figures/lyap_invariant.tex | 25 + figures/lyapunov_stamp.jpg | Bin 0 -> 45637 bytes figures/mass-damper.tex | 41 + figures/pend_lyap.pdf | Bin 0 -> 284224 bytes figures/pend_lyap.png | Bin 0 -> 764542 bytes figures/pend_lyap.tex | 10 + figures/pend_lyap_eps.eps | 96421 +++++++++++++++++++++++++++++++++++ 15 files changed, 105461 insertions(+), 88 deletions(-) create mode 100644 .gitignore create mode 100644 beamerExample.tex create mode 100644 figures/count_cont.eps create mode 100644 figures/count_cont.png create mode 100644 figures/guck.eps create mode 100644 figures/lyap.tex create mode 100644 figures/lyap_ex.tex create mode 100644 figures/lyap_invariant.tex create mode 100644 figures/lyapunov_stamp.jpg create mode 100644 figures/mass-damper.tex create mode 100644 figures/pend_lyap.pdf create mode 100644 figures/pend_lyap.png create mode 100644 figures/pend_lyap.tex create mode 100644 figures/pend_lyap_eps.eps diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..77449df --- /dev/null +++ b/.gitignore @@ -0,0 +1,307 @@ +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Generated if empty string is given at "Please type another file name for output:" +.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# svg +svg-inkscape/ + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# titletoc +*.ptc + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# xwatermark package +*.xwm + +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib +# option is specified. Footnotes are the stored in a file with suffix Notes.bib. +# Uncomment the next line to have this generated file ignored. +#*Notes.bib + +# Emacs backup files +*~ + +# build directories +build/ \ No newline at end of file diff --git a/README.md b/README.md index e94c39c..5c07188 100644 --- a/README.md +++ b/README.md @@ -1,92 +1,5 @@ # beamerExample +This repo contains a complete example for `\usetheme{Regler}` -## Getting started - -To make it easy for you to get started with GitLab, here's a list of recommended next steps. - -Already a pro? Just edit this README.md and make it your own. Want to make it easy? [Use the template at the bottom](#editing-this-readme)! - -## Add your files - -- [ ] [Create](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#create-a-file) or [upload](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#upload-a-file) files -- [ ] [Add files using the command line](https://docs.gitlab.com/ee/gitlab-basics/add-file.html#add-a-file-using-the-command-line) or push an existing Git repository with the following command: - -``` -cd existing_repo -git remote add origin https://gitlab.control.lth.se/regler/beamerExample.git -git branch -M main -git push -uf origin main -``` - -## Integrate with your tools - -- [ ] [Set up project integrations](https://gitlab.control.lth.se/regler/beamerExample/-/settings/integrations) - -## Collaborate with your team - -- [ ] [Invite team members and collaborators](https://docs.gitlab.com/ee/user/project/members/) -- [ ] [Create a new merge request](https://docs.gitlab.com/ee/user/project/merge_requests/creating_merge_requests.html) -- [ ] [Automatically close issues from merge requests](https://docs.gitlab.com/ee/user/project/issues/managing_issues.html#closing-issues-automatically) -- [ ] [Enable merge request approvals](https://docs.gitlab.com/ee/user/project/merge_requests/approvals/) -- [ ] [Automatically merge when pipeline succeeds](https://docs.gitlab.com/ee/user/project/merge_requests/merge_when_pipeline_succeeds.html) - -## Test and Deploy - -Use the built-in continuous integration in GitLab. - -- [ ] [Get started with GitLab CI/CD](https://docs.gitlab.com/ee/ci/quick_start/index.html) -- [ ] [Analyze your code for known vulnerabilities with Static Application Security Testing(SAST)](https://docs.gitlab.com/ee/user/application_security/sast/) -- [ ] [Deploy to Kubernetes, Amazon EC2, or Amazon ECS using Auto Deploy](https://docs.gitlab.com/ee/topics/autodevops/requirements.html) -- [ ] [Use pull-based deployments for improved Kubernetes management](https://docs.gitlab.com/ee/user/clusters/agent/) -- [ ] [Set up protected environments](https://docs.gitlab.com/ee/ci/environments/protected_environments.html) - -*** - -# Editing this README - -When you're ready to make this README your own, just edit this file and use the handy template below (or feel free to structure it however you want - this is just a starting point!). Thank you to [makeareadme.com](https://www.makeareadme.com/) for this template. - -## Suggestions for a good README -Every project is different, so consider which of these sections apply to yours. The sections used in the template are suggestions for most open source projects. Also keep in mind that while a README can be too long and detailed, too long is better than too short. If you think your README is too long, consider utilizing another form of documentation rather than cutting out information. - -## Name -Choose a self-explaining name for your project. - -## Description -Let people know what your project can do specifically. Provide context and add a link to any reference visitors might be unfamiliar with. A list of Features or a Background subsection can also be added here. If there are alternatives to your project, this is a good place to list differentiating factors. - -## Badges -On some READMEs, you may see small images that convey metadata, such as whether or not all the tests are passing for the project. You can use Shields to add some to your README. Many services also have instructions for adding a badge. - -## Visuals -Depending on what you are making, it can be a good idea to include screenshots or even a video (you'll frequently see GIFs rather than actual videos). Tools like ttygif can help, but check out Asciinema for a more sophisticated method. - -## Installation -Within a particular ecosystem, there may be a common way of installing things, such as using Yarn, NuGet, or Homebrew. However, consider the possibility that whoever is reading your README is a novice and would like more guidance. Listing specific steps helps remove ambiguity and gets people to using your project as quickly as possible. If it only runs in a specific context like a particular programming language version or operating system or has dependencies that have to be installed manually, also add a Requirements subsection. - -## Usage -Use examples liberally, and show the expected output if you can. It's helpful to have inline the smallest example of usage that you can demonstrate, while providing links to more sophisticated examples if they are too long to reasonably include in the README. - -## Support -Tell people where they can go to for help. It can be any combination of an issue tracker, a chat room, an email address, etc. - -## Roadmap -If you have ideas for releases in the future, it is a good idea to list them in the README. - -## Contributing -State if you are open to contributions and what your requirements are for accepting them. - -For people who want to make changes to your project, it's helpful to have some documentation on how to get started. Perhaps there is a script that they should run or some environment variables that they need to set. Make these steps explicit. These instructions could also be useful to your future self. - -You can also document commands to lint the code or run tests. These steps help to ensure high code quality and reduce the likelihood that the changes inadvertently break something. Having instructions for running tests is especially helpful if it requires external setup, such as starting a Selenium server for testing in a browser. - -## Authors and acknowledgment -Show your appreciation to those who have contributed to the project. - -## License -For open source projects, say how it is licensed. - -## Project status -If you have run out of energy or time for your project, put a note at the top of the README saying that development has slowed down or stopped completely. Someone may choose to fork your project or volunteer to step in as a maintainer or owner, allowing your project to keep going. You can also make an explicit request for maintainers. diff --git a/beamerExample.tex b/beamerExample.tex new file mode 100644 index 0000000..f75d09e --- /dev/null +++ b/beamerExample.tex @@ -0,0 +1,603 @@ +\documentclass{beamer} +\usetheme{Regler} +\usepackage{standalone} +\usepackage{tikz} +\usetikzlibrary{shapes,positioning} +\usetikzlibrary{narrow} +\usetikzlibrary{calc,patterns,decorations.pathmorphing,decorations.markings} + +\begin{document} +\title[Nonlinear Control and Servo Systems]{Nonlinear Control\\ and Servo Systems} +\subtitle{Lecture 2} +\author{Anders Robertsson} +\institute{Dept. of Automatic Control\\Lund Institute of Technology} +\maketitle + +% ---------------------------------------------------------------------- + +\begin{frame}{Today's Goal} + To be able to + \begin{itemize}\slshape + \item prove local and global stability of an equilibrium point + through Lyapunov's method \pause + \item show stability of a set (for example, a limit cycle) through + invariant set theorems + \end{itemize} +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame}{Material} + \begin{itemize} + \item Slotine and Li: Chapter~3 + \item Lecture notes + \end{itemize} +\end{frame} + + +% ---------------------------------------------------------------------- +\begin{frame}{\hspace*{10pt}Alexandr Mihailovich Lyapunov (1857--1918)} +\begin{columns} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{figures/lyapunov_stamp} + \end{column} +\begin{column}{0.7\textwidth} +\pause +\begin{block}{Master's thesis} + ``On the stability of ellipsoidal forms of equilibrium of + rotating fluids,'' St. Petersburg University, 1884. +\end{block} +\pause +\begin{block}{Doctoral thesis} + ``The general problem of the stability of motion,'' 1892. +\end{block} +\end{column} +\end{columns} +\end{frame} +% ---------------------------------------------------------------------- + +% ---------------------------------------------------------------------- + +\begin{frame}{Lyapunov's idea} + + \begin{center} + \textbf{\emph{If the total energy is dissipated, the system must be + stable.}} + \end{center} + \pause\setbeamertemplate{navigation symbols}[horizontal] + \begin{block}{Main benefit} + By looking at an energy-like function (a so called Lyapunov + function), we might conclude that a system is stable or + asymptotically stable \textbf{without solving} the nonlinear + differential equation. + \end{block} + \pause \medskip + \begin{block}{Main question} How to find a Lyapunov function? + \end{block} +\end{frame} + +% ---------------------------------------------------------------------- +\begin{frame}{A Motivating Example} + + \begin{centering} + \input{figures/mass-damper}\par + \end{centering} + + \vspace*{-6mm} + \begin{equation*} + m\ddot{x}=-\underbrace{b\dot{x}\vert\dot{x}\vert}_{\text{damping}} + -\underbrace{k_0x-k_1x^3}_{\text{spring}},\qquad b,k_0,k_1>0 + \end{equation*} + + The energy can be shown to be + \begin{equation*} + \begin{aligned} + V(x,\dot{x})&=m\dot{x}^2/2+k_0x^2/2+k_1x^4/4>0, & V(0,0)&=0\\ + % + \frac{d}{dt}{V}(x,\dot{x})&=m\dot{x}\ddot{x}+k_0x\dot{x}+k_1x^3\dot{x} + =-b\vert\dot{x}\vert^3<0, &\dot{x}&\neq 0 + \end{aligned} + \end{equation*} + +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame}{Stability Definitions} + + An equilibrium point $x=0$ of $\dot{x}=f(x)$ is + + \textbf{locally stable}, if for every $R>0$ there exists $r>0$, such + that + \begin{equation*} + \Vert x(0)\Vert<r \quad\Rightarrow\quad \Vert x(t)\Vert<R, + \quad t\geq 0 + \end{equation*} + \pause + \textbf{locally asymptotically stable}, if locally stable and + \begin{equation*} + \Vert x(0)\Vert<r \quad\Rightarrow\quad \lim_{t\rightarrow\infty}x(t)=0 + \end{equation*} + \pause + \textbf{globally asymptotically stable}, if asymptotically stable + for all $x(0)\in\mathbf{R}^n$. +\end{frame} +%%%%---------------------------------------------------------------------- + +\begin{frame} +\frametitle{Lyapunov Theorem for Local Stability} +\begin{block}{Theorem} + Let $\dot{x}=f(x)$, $f(0)=0$, and $0\in\Omega\subset\mathbf{R}^n$. + +Assume that $V:\Omega\rightarrow\mathbf{R}$ is a $C^1$ function. If +\begin{itemize} +\item $V(0)=0$ +\item $V(x)>0$, for all $x\in\Omega$, $x\neq 0$ +\item $\dfrac{d}{dt}{V}(x)\leq 0$ along all trajectories in $\Omega$ +\end{itemize} +then $x=0$ is locally stable. Furthermore, if also +\begin{itemize} +\item $\dfrac{d}{dt}{V}(x)<0$ for all $x\in\Omega$, $x\neq 0$ +\end{itemize} +then $x=0$ is locally asymptotically stable. +\end{block} +\vfill +Proof: see p.~62. +\end{frame} +% ---------------------------------------------------------------------- + +\begin{frame}{Lyapunov Functions ($\approx$ Energy Functions)} + +A \emph{Lyapunov function} fulfills $V(x_0)=0$, $V(x)>0$ for $x\in\Omega$, +$x\neq x_0$, and +\begin{equation*} +\dot{V}(x)=\frac{d}{dt}V(x)=\frac{dV}{dx}\dot{x}=\frac{dV}{dx}f(x)\leq 0 +\end{equation*} +\medskip +\begin{columns} +\begin{column}{0.59\hsize}\centering +\input{figures/lyap} +\end{column} +\begin{column}{0.39\hsize}\centering +\input{figures/lyap_ex} +\end{column} +\end{columns} +\end{frame} + + +% ---------------------------------------------------------------------- +\begin{frame} +\frametitle{2 min exercise---Pendulum} + +Show that the origin is locally stable for a mathematical pendulum. +\begin{equation*} + \dot{x}_1=x_2, \quad \dot{x}_2=-\frac{g}{\ell}\sin x_1 +\end{equation*} +Use as a Lyapunov function candidate +\begin{equation*} +V(x)=(1-\cos x_1)g\ell+\ell^2x_2^2/2 +\end{equation*} +\hrule height 0pt +\vspace*{-3mm} + \begin{centering} + \includegraphics[width=0.5\hsize]{figures/pend_lyap}\par + \end{centering} +\end{frame} + +% ---------------------------------------------------------------------- +% +% \begin{frame} +% \frametitle{Example---Pendulum} +% \begin{itemize} +% \item $V(0)=0$ +% \item $V(x)>0$ for $-2\pi<x_1<2\pi$ and $(x_1,x_2)\neq 0$ +% \item +% $$ +% \dot{V}(x)=\frac{g}{\ell}\dot{x}_1\sin x_1+x_2\dot{x}_2=0, +% \qquad\text{for all $x$} +% $$ +% \end{itemize} +% Hence, $x=0$ is locally stable. +% +% $x=0$ is not asymptotically stable, because +% $\dot{V}(x)=0$ for all $x$. +% } % \end{slide} +% ---------------------------------------------------------------------- + +% +\begin{frame} + \frametitle{Lyapunov Theorem for Global Stability} + +\textbf{Theorem} +Let $\dot{x}=f(x)$ and $f(0)=0$. +Assume that $V:\mathbf{R}^n\rightarrow\mathbf{R}$ is a $C^1$ function. If +\begin{itemize} +\item $V(0)=0$ +\item $V(x)>0$, for all $x\neq 0$ +\item $\dot{V}(x)< 0$ for all $x\neq 0$ +\item $V(x)\rightarrow\infty$ as $\Vert x\Vert\rightarrow\infty$ +\end{itemize} +then $x=0$ is globally asymptotically stable. + +\end{frame} + + +% ---------------------------------------------------------------------- + +% +\begin{frame} \frametitle{Radial Unboundedness is Necessary} + +If the condition $V(x)\rightarrow\infty$ as $\Vert +x\Vert\rightarrow\infty$ is not fulfilled, then global stability +cannot be guaranteed. + +\textbf{Example} Assume $V(x)=x_1^2/(1+x_1^2)+x_2^2$ +is a Lyapunov function for a system. Can have $\Vert x\Vert\rightarrow\infty$ +even if $\dot{V}(x)<0$. + +\medskip +\centerline{Contour plot $V(x)=C$:} +\begin{center} +% \psfrag{x1}[][]{$x_1$} +% \psfrag{x2}[][]{$x_2$} +\includegraphics[width=0.5\hsize]{figures/count_cont} +\end{center} +\end{frame} + + + +% ---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Positive Definite Matrices} + +A matrix $M$ is \textbf{positive definite} if $x^TMx>0$ for all $x\neq +0$.\\ + It is \textbf{positive semidefinite} if $x^TMx\geq 0$ for +all $x$. + +A symmetric matrix $M=M^T$ is positive definite if and only if its +eigenvalues $\lambda_i>0$. (\emph{semidefinite} $\Leftrightarrow$ +$\lambda_i\geq 0$) + +Note that if $M=M^T$ is positive definite, then the Lyapunov function +candidate $V(x)=x^TMx$ fulfills $V(0)=0$ and $V(x)>0$ for all $x\neq +0$. +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{More matrix results} + +A symmetric matrix $M=M^T$ satisfies the inequalities +$$ + \lambda_{\min}(M)\Vert x\Vert^2\enskip \leq \enskip x^TMx \enskip +\leq \enskip \lambda_{\max}(M)\Vert x\Vert^2 +$$ + +(To show it, use the factorization $M=U\Lambda U^*$, where $U$ is a +unitary matrix, $\Vert +Ux\Vert=\Vert x\Vert$, $U^*$ is complex conjugate transpose, and +$\Lambda=\text{diag}(\lambda_1,\dots,\lambda_n)$.) + +For any matrix $M$ one also has +\[ +\Vert Mx \Vert \enskip \leq \enskip \lambda^{1/2}_{\max}(M^TM) \Vert x \Vert +\] +% Horn & Johnsson p 176 +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Lyapunov Function for Linear System} + +\textbf{Theorem} +The eigenvalues $\lambda_i$ of $A$ satisfy +$\text{Re} \,\lambda_i<0$ if and only if: for every positive definite +$Q=Q^T$ there exists a positive definite $P=P^T$ such that +$$ +PA+A^TP=-Q +$$ +\emph{Proof of \underline{$\exists Q,P \Rightarrow \text{Re} \,\lambda_i(A)<0$}:} +Consider $\dot{x}=Ax$ and the Lyapunov function +candidate $V(x)=x^TPx$. +$$ +\dot{V}(x)=x^TP\dot{x}+\dot{x}^TPx=x^T(PA+A^TP)x=-x^TQx<0, \quad +\forall x\neq 0 +$$ +$$ +\Rightarrow\quad \dot{x}=Ax \quad\text{asymptotically + stable}\quad\Longleftrightarrow\qquad \text{Re}\, \lambda_i<0 +$$ +\emph{Proof of \underline{$\text{Re}\, + \lambda_i(A)<0 \Rightarrow \exists Q,P$:}} +Choose $P=\int_0^{\infty}e^{A^Tt}Qe^{At}dt$ +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Lyapunov's Linearization Method} + +Recall from Lecture~2: + +\textbf{Theorem} +Consider +$$ +\dot{x}=f(x) +$$ +Assume that $x=0$ is an equilibrium point and that +$$ +\dot{x}=Ax+g(x) +$$ +is a linearization. +\begin{itemize} +\item[(1)] If $\text{Re}\,\lambda_i(A)<0$ for all $i$, then $x=0$ is locally + asymptotically stable. +\item[(2)] If there exists $i$ such that $\lambda_i(A)>0$, +then $x=0$ is unstable. +\end{itemize} +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{\large Proof of (1) in Lyapunov's Linearization Method} + +Lyapunov function candidate $V(x)=x^TPx$. $V(0)=0$, $V(x)>0$ for +$x\neq 0$, and +\begin{align*} +\dot{V}(x)&=x^TPf(x)+f^T(x)Px\\ +&=x^TP[Ax+g(x)]+[x^TA+g^T(x)]Px\\ +&=x^T(PA+A^TP)x+2x^TPg(x)=-x^TQx+2x^TPg(x) +\end{align*} +% +\begin{equation*} +x^TQx\geq\lambda_{\min}(Q)\Vert x\Vert^2 +\end{equation*} +% +and for all $\gamma>0$ there exists $r>0$ such that +% +\begin{equation*} +\Vert g(x)\Vert<\gamma\Vert x\Vert, \qquad \forall\Vert x\Vert<r +\end{equation*} +% +Thus, choosing $\gamma$ sufficiently small gives +% +\begin{equation*} +\dot{V}(x)\leq-\big(\lambda_{\min}(Q)-2\gamma\lambda_{\max}(P)\big)\Vert x\Vert^2<0 +\end{equation*} +% +\end{frame} +% ---------------------------------------------------------------------- + +\end{document} + + +\begin{frame} +% +\frametitle{Invariant Sets} + + +\textbf{Definition} A set $M$ is called \textbf{invariant} if for +the system +$$ +\dot{x}=f(x), +$$ +$x(0)\in M$ implies that $x(t)\in M$ for all $t\geq 0$. + +\medskip +\begin{center} + \psfrag{x0}[][]{$x(0)$} + \psfrag{xt}[][]{$x(t)$} + \psfrag{M}[][]{$M$} +\includegraphics[width=0.5\hsize]{figures/invariant_set.eps} +\end{center} +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Invariant Set Theorem} + +\textbf{Theorem} Let $\Omega\in\mathbf{R}^n$ be a bounded and closed set +that is invariant with respect to +$$ +\dot{x}=f(x). +$$ +Let $V:\mathbf{R}^n\rightarrow\mathbf{R}$ be a radially unbounded +$C^1$ function such that +$\dot{V}(x)\leq 0$ for $x\in\Omega$. Let $E$ be the set of points +in $\Omega$ where $\dot{V}(x)=0$. If $M$ is the largest invariant set in +$E$, then every solution with $x(0)\in\Omega$ approaches $M$ as +$t\rightarrow\infty$ (proof on p.~73) + +\begin{center} + \psfrag{Om}[][]{$\Omega$} + \psfrag{E}[][]{$E$} + \psfrag{M}[][]{$M$} +\includegraphics[width=0.4\hsize]{figures/lyap_invariant.eps} +\end{center} +\end{frame} +%---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Example---Stable Limit Cycle} + +Show that $M=\{x:\,\Vert x\Vert= 1\}$ is a globally stable limit cycle for +\begin{align*} + \dot{x}_1&=x_1-x_2-x_1(x_1^2+x_2^2)\\ + \dot{x}_2&=x_1+x_2-x_2(x_1^2+x_2^2) +\end{align*} + +Let $V(x)=(x_1^2+x_2^2-1)^2$. +\begin{align*} +\frac{dV}{dt}&=2(x_1^2+x_2^2-1)\frac{d}{dt}(x_1^2+x_2^2-1)\\ +&=-2(x_1^2+x_2^2-1)^2(x_1^2+x_2^2)\leq 0\quad\text{for } x\in\Omega\\ +\end{align*} +$\Omega=\{0<\Vert x\Vert\leq R\}$ is invariant for $R=1$. +\end{frame} +%---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Example---Stable Limit Cycle} + +\begin{equation*} +E=\{x\in\Omega:\,\dot{V}(x)=0\}=\{x:\,\Vert x\Vert= 1\} +\end{equation*} +$M=E$ is an invariant set, because +\begin{equation*} +\frac{d}{dt}V=-2(x_1^2+x_2^2-1)(x_1^2+x_2^2)=0\quad\text{for + } x\in M +\end{equation*} + +We have shown that $M$ is a stable limit cycle (globally stable in +$R-\{0\}$) + \begin{center} + \includegraphics[height=0.3\hsize]{figures/guck} + \end{center} + +\end{frame} +% ---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{A Motivating Example (cont'd)} + +$$ +m\ddot{x}=-b\dot{x}\vert\dot{x}\vert-k_0x-k_1x^3 +$$ +$$ +V(x,\dot{x})=(2m\dot{x}^2+2k_0x^2+k_1x^4)/4>0, \qquad V(0,0)=0 +$$ +$\dot{V}(x,\dot{x})=-b\vert\dot{x}\vert^3$ gives +$E=\{(x,\dot{x}):\,\dot{x}=0\}$. + +Assume there exists $(\bar{x},\dot{\bar{x}})\in M$ such that +$\bar{x}(t_0)\neq 0$. Then +$$ +m\ddot{\bar{x}}(t_0)=-k_0\bar{x}(t_0)-k_1\bar{x}^3(t_0)\neq 0 +$$ +so $\dot{\bar{x}}(t_0+)\neq 0$ so the trajectory will immediately +leave $M$. A contradiction to that $M$ is invariant. + +Hence, $M=\{(0,0)\}$ so the origin is asymptotically stable. +\end{frame} + +%---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Adaptive Noise Cancellation by Lyapunov Design} + + \begin{center} + \psfrag{u}[][]{$u$} + \psfrag{g1}[][][1.4]{$\frac{b}{s+a}$} + \psfrag{g2}[][][1.4]{$\frac{\widehat b}{s+\widehat a}$} + \psfrag{x}[][]{$x$} + \psfrag{xh}[][]{$\widehat x$} + \psfrag{xt}[][]{$\widetilde x$} + \psfrag{+}[][]{$+$} + \psfrag{-}[][]{$-$} + \includegraphics[width=0.4\hsize]{figures/noise.eps} + \end{center} +\begin{align*} +\dot x +a x &= bu\\ +\dot{\widehat x} + \widehat a \widehat x &= \widehat b u +\end{align*} +Introduce $\widetilde x = x-\widehat x, \enskip\widetilde a = +a-\widehat a, \enskip \widetilde b = b-\widehat b$. + +Want to design adaptation law so that $\widetilde x\to 0$ +\end{frame} + +\begin{frame} +Let us try the Lyapunov function +\begin{align*} +V&=\frac{1}{2}(\widetilde x^2+\gamma_a\widetilde a^2+\gamma_b\widetilde +b^2)\\ +\dot V &= \widetilde x \dot{\widetilde x} + \gamma_a \widetilde a +\dot{\widetilde a} + \gamma_b \widetilde b +\dot{\widetilde b} = \\ +&=\widetilde x(-a\widetilde x-\widetilde a \widehat x + \widetilde b +u) + \gamma_a \widetilde a +\dot{\widetilde a} + \gamma_b \widetilde b +\dot{\widetilde b} = -a \widetilde x^2 +\end{align*} +where the last equality follows if we choose +\begin{align*} +\dot{\widetilde a} = -\dot{\widehat a} = \frac{1}{\gamma_a} \widetilde{x} \widehat x \qquad +\dot{\widetilde b} = -\dot{\widehat b} = -\frac{1}{\gamma_b} \widetilde{x} u +\end{align*} +Invariant set: $\widetilde x =0$. + +This proves that $\widetilde x \to 0$. + +(The parameters $\widetilde a$ and $\widetilde b$ +do not necessarily converge: $u\equiv 0$.) + +\fbox{Demonstration if time permits} +\end{frame} +%%------------------------------ +\begin{frame} +\frametitle{Results} + \begin{center} + \psfrag{p1}[][][1.3]{$\hat{a}$} + \psfrag{p2}[][][1.3]{$\hat{b}$} + \includegraphics[width=0.7\hsize]{musik/parameters.eps} \\ + Estimation of parameters starts at t=10 s. + \end{center} +\end{frame} + +%---------------------------------------------------------------------- +\begin{frame} +\frametitle{Results} + \begin{center} + \psfrag{x-xhat}[][][1.3]{$x-\hat{x}$} + \includegraphics[width=0.6\hsize,angle=0]{musik/adap.ps} \\ + \includegraphics[width=0.3\hsize]{musik/xt.eps} + \end{center} + \begin{center} + \begin{small} + Estimation of parameters starts at t=10 s. + \end{small} + \end{center} +\end{frame} + +%---------------------------------------------------------------------- +\begin{frame} +% +\frametitle{Next Lecture} + + +\begin{center} + Stability analysis using input-output (frequency) methods + +\end{center} +\begin{itemize} +item Stability analysis using input-output (frequency) methods +\end{itemize} + +\vfill +\end{frame} +%---------------------------------------------------------------------- +\begin{frame} +\frametitle{title} + Let us try the Lyapunov function +\begin{align*} + V&=\frac{1}{2}(\widetilde x^2+\gamma_a\widetilde a^2+\gamma_b\widetilde + b^2)\\ + \dot V &= \widetilde x \dot{\widetilde x} + \gamma_a \widetilde a + \dot{\widetilde a} + \gamma_b \widetilde b + \dot{\widetilde b} = \\ + &=\widetilde x(-a\widetilde x-\widetilde a \widehat x + \widetilde b + u) + \gamma_a \widetilde a + \dot{\widetilde a} + \gamma_b \widetilde b + \dot{\widetilde b} = -a \widetilde x^2 +\end{align*} + where the last equality follows if we choose + \begin{align*} + \dot{\widetilde a} = -\dot{\widehat a} = \frac{1}{\gamma_a} \widetilde{x} \widehat x \qquad + \dot{\widetilde b} = -\dot{\widehat b} = -\frac{1}{\gamma_b} \widetilde{x} u + \end{align*} + Invariant set: $\widetilde x =0$. + + This proves that $\widetilde x \to 0$. + + (The parameters $\widetilde a$ and $\widetilde b$ + do not necessarily converge: $u\equiv 0$.) + + \fbox{Demonstration if time permits} + +\end{frame} \ No newline at end of file diff --git a/figures/count_cont.eps b/figures/count_cont.eps new file mode 100644 index 0000000..356d956 --- /dev/null +++ b/figures/count_cont.eps @@ -0,0 +1,4648 @@ +%!PS-Adobe-2.0 +%%Creator: MATLAB, The Mathworks, Inc. +%%Title: count_cont.ps +%%CreationDate: 01/16/98 12:18:15 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Pages: (atend) +%%BoundingBox: (atend) +%%EndComments + +%%BeginProlog + +% MathWorks dictionary +/MathWorks 160 dict begin + +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef + +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +/rm /rmoveto ldef +/rl /rlineto 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+1918 1383 mt 1944 1382 L +1944 1382 mt 1971 1382 L +1971 1382 mt 1998 1381 L +1998 1381 mt 2025 1381 L +2025 1381 mt 2052 1380 L +2052 1380 mt 2078 1380 L +2078 1380 mt 2105 1379 L +2105 1379 mt 2132 1379 L +2132 1379 mt 2159 1378 L +2159 1378 mt 2186 1377 L +2186 1377 mt 2212 1377 L +2212 1377 mt 2239 1376 L +2239 1376 mt 2266 1375 L +2266 1375 mt 2293 1374 L +2293 1374 mt 2320 1374 L +2320 1374 mt 2346 1373 L +2346 1373 mt 2373 1372 L +2373 1372 mt 2400 1371 L +2400 1371 mt 2427 1370 L +2427 1370 mt 2454 1368 L +2454 1368 mt 2480 1367 L +2480 1367 mt 2507 1366 L +2507 1366 mt 2534 1364 L +2534 1364 mt 2561 1363 L +2561 1363 mt 2588 1361 L +2588 1361 mt 2615 1359 L +2615 1359 mt 2641 1357 L +2641 1357 mt 2668 1355 L +2668 1355 mt 2695 1353 L +2695 1353 mt 2722 1351 L +2722 1351 mt 2749 1348 L +2749 1348 mt 2775 1345 L +2775 1345 mt 2802 1342 L +2802 1342 mt 2817 1340 L +2817 1340 mt 2829 1339 L +2829 1339 mt 2856 1335 L +2856 1335 mt 2883 1331 L +2883 1331 mt 2909 1327 L +2909 1327 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1041 mt 1462 1040 L +1462 1040 mt 1489 1040 L +1489 1040 mt 1516 1040 L +1516 1040 mt 1542 1040 L +1542 1040 mt 1569 1040 L +1569 1040 mt 1596 1040 L +1596 1040 mt 1623 1039 L +1623 1039 mt 1650 1039 L +1650 1039 mt 1676 1039 L +1676 1039 mt 1703 1039 L +1703 1039 mt 1730 1038 L +1730 1038 mt 1757 1038 L +1757 1038 mt 1784 1038 L +1784 1038 mt 1810 1038 L +1810 1038 mt 1837 1037 L +1837 1037 mt 1864 1037 L +1864 1037 mt 1891 1037 L +1891 1037 mt 1918 1037 L +1918 1037 mt 1944 1036 L +1944 1036 mt 1971 1036 L +1971 1036 mt 1998 1036 L +1998 1036 mt 2025 1035 L +2025 1035 mt 2052 1035 L +2052 1035 mt 2078 1034 L +2078 1034 mt 2105 1034 L +2105 1034 mt 2132 1034 L +2132 1034 mt 2159 1033 L +2159 1033 mt 2186 1033 L +2186 1033 mt 2212 1032 L +2212 1032 mt 2239 1031 L +2239 1031 mt 2266 1031 L +2266 1031 mt 2293 1030 L +2293 1030 mt 2320 1030 L +2320 1030 mt 2346 1029 L +2346 1029 mt 2373 1028 L +2373 1028 mt 2400 1027 L +2400 1027 mt 2427 1026 L +2427 1026 mt 2454 1026 L +2454 1026 mt 2480 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+3472 747 mt 3499 729 L +3499 729 mt 3526 715 L +3526 715 mt 3553 706 L +3553 706 mt 3553 706 L +3553 706 mt 3580 703 L +3580 703 mt 3606 706 L +3606 706 mt 3606 706 L +3606 706 mt 3633 715 L +3633 715 mt 3660 729 L +3660 729 mt 3687 747 L +3687 747 mt 3714 767 L +3714 767 mt 3740 788 L +3740 788 mt 3767 808 L +3767 808 mt 3772 812 L +3772 812 mt 3794 829 L +3794 829 mt 3821 848 L +3821 848 mt 3848 866 L +3848 866 mt 3874 882 L +3874 882 mt 3901 897 L +3901 897 mt 3928 910 L +3928 910 mt 3945 917 L +3945 917 mt 3955 922 L +3955 922 mt 3982 933 L +3982 933 mt 4008 943 L +4008 943 mt 4035 951 L +4035 951 mt 4062 959 L +4062 959 mt 4089 966 L +4089 966 mt 4116 972 L +4116 972 mt 4142 978 L +4142 978 mt 4169 983 L +4169 983 mt 4196 987 L +4196 987 mt 4223 991 L +4223 991 mt 4250 995 L +4250 995 mt 4276 998 L +4276 998 mt 4303 1001 L +4303 1001 mt 4330 1004 L +4330 1004 mt 4357 1006 L +4357 1006 mt 4384 1008 L +4384 1008 mt 4410 1011 L +4410 1011 mt 4437 1012 L +4437 1012 mt 4464 1014 L 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L +5751 522 mt 5778 522 L +5778 522 mt 5804 522 L +5804 522 mt 5831 522 L +5831 522 mt 5858 522 L +5858 522 mt 5885 522 L +5885 522 mt 5912 522 L +5912 522 mt 5938 522 L +5938 522 mt 5965 523 L +5965 523 mt 5992 523 L +5992 523 mt 6019 523 L +6019 523 mt 6046 523 L +6046 523 mt 6072 523 L +6072 523 mt 6099 523 L +6099 523 mt 6126 523 L +6126 523 mt 6153 523 L +6153 523 mt 6180 523 L +6180 523 mt 6206 523 L +6206 523 mt 6233 523 L +6233 523 mt 6260 523 L +3848 4616 mt 3848 4616 L +3848 4616 mt 3874 4603 L +3874 4603 mt 3901 4592 L +3901 4592 mt 3928 4581 L +3928 4581 mt 3955 4572 L +3955 4572 mt 3982 4564 L +3982 4564 mt 4008 4557 L +4008 4557 mt 4035 4550 L +4035 4550 mt 4062 4544 L +4062 4544 mt 4089 4539 L +4089 4539 mt 4116 4535 L +4116 4535 mt 4142 4531 L +4142 4531 mt 4169 4527 L +4169 4527 mt 4196 4524 L +4196 4524 mt 4223 4521 L +4223 4521 mt 4250 4518 L +4250 4518 mt 4276 4515 L +4276 4515 mt 4303 4513 L +4303 4513 mt 4330 4511 L +4330 4511 mt 4342 4510 L +4342 4510 mt 4357 4509 L +4357 4509 mt 4384 4507 L +4384 4507 mt 4410 4506 L +4410 4506 mt 4437 4504 L +4437 4504 mt 4464 4503 L +4464 4503 mt 4491 4502 L +4491 4502 mt 4518 4500 L +4518 4500 mt 4544 4499 L +4544 4499 mt 4571 4498 L +4571 4498 mt 4598 4497 L +4598 4497 mt 4625 4497 L +4625 4497 mt 4652 4496 L +4652 4496 mt 4679 4495 L +4679 4495 mt 4705 4494 L +4705 4494 mt 4732 4494 L +4732 4494 mt 4759 4493 L +4759 4493 mt 4786 4492 L +4786 4492 mt 4813 4492 L +4813 4492 mt 4839 4491 L +4839 4491 mt 4866 4491 L +4866 4491 mt 4893 4490 L +4893 4490 mt 4920 4490 L +4920 4490 mt 4947 4489 L +4947 4489 mt 4973 4489 L +4973 4489 mt 5000 4489 L +5000 4489 mt 5027 4488 L +5027 4488 mt 5054 4488 L +5054 4488 mt 5081 4488 L +5081 4488 mt 5107 4487 L +5107 4487 mt 5134 4487 L +5134 4487 mt 5161 4487 L +5161 4487 mt 5188 4487 L +5188 4487 mt 5215 4486 L +5215 4486 mt 5241 4486 L +5241 4486 mt 5268 4486 L +5268 4486 mt 5295 4486 L +5295 4486 mt 5322 4486 L +5322 4486 mt 5349 4485 L +5349 4485 mt 5375 4485 L +5375 4485 mt 5402 4485 L +5402 4485 mt 5429 4485 L +5429 4485 mt 5456 4485 L +5456 4485 mt 5483 4484 L +5483 4484 mt 5509 4484 L +5509 4484 mt 5536 4484 L +5536 4484 mt 5563 4484 L +5563 4484 mt 5590 4484 L +5590 4484 mt 5617 4484 L +5617 4484 mt 5643 4484 L +5643 4484 mt 5670 4484 L +5670 4484 mt 5697 4483 L +5697 4483 mt 5724 4483 L +5724 4483 mt 5751 4483 L +5751 4483 mt 5778 4483 L +5778 4483 mt 5804 4483 L +5804 4483 mt 5831 4483 L +5831 4483 mt 5858 4483 L +5858 4483 mt 5885 4483 L +5885 4483 mt 5912 4483 L +5912 4483 mt 5938 4483 L +5938 4483 mt 5965 4482 L +5965 4482 mt 5992 4482 L +5992 4482 mt 6019 4482 L +6019 4482 mt 6046 4482 L +6046 4482 mt 6072 4482 L +6072 4482 mt 6099 4482 L +6099 4482 mt 6126 4482 L +6126 4482 mt 6153 4482 L +6153 4482 mt 6180 4482 L +6180 4482 mt 6206 4482 L +6206 4482 mt 6233 4482 L +6233 4482 mt 6260 4482 L +2775 389 mt 2775 389 L +2775 389 mt 2749 391 L +2749 391 mt 2722 392 L +2722 392 mt 2695 393 L +2695 393 mt 2668 395 L +2668 395 mt 2641 396 L +2641 396 mt 2615 397 L +2615 397 mt 2588 398 L +2588 398 mt 2561 399 L +2561 399 mt 2534 399 L +2534 399 mt 2507 400 L +2507 400 mt 2480 401 L +2480 401 mt 2454 402 L +2454 402 mt 2427 402 L +2427 402 mt 2400 403 L +2400 403 mt 2373 403 L +2373 403 mt 2346 404 L +2346 404 mt 2320 404 L +2320 404 mt 2293 405 L +2293 405 mt 2266 405 L +2266 405 mt 2239 406 L +2239 406 mt 2212 406 L +2212 406 mt 2186 406 L +2186 406 mt 2159 407 L +2159 407 mt 2132 407 L +2132 407 mt 2105 407 L +2105 407 mt 2078 408 L +2078 408 mt 2052 408 L +2052 408 mt 2025 408 L +2025 408 mt 1998 409 L +1998 409 mt 1971 409 L +1971 409 mt 1944 409 L +1944 409 mt 1918 409 L +1918 409 mt 1891 409 L +1891 409 mt 1864 410 L +1864 410 mt 1837 410 L +1837 410 mt 1810 410 L +1810 410 mt 1784 410 L +1784 410 mt 1757 410 L +1757 410 mt 1730 411 L +1730 411 mt 1703 411 L +1703 411 mt 1676 411 L +1676 411 mt 1650 411 L +1650 411 mt 1623 411 L +1623 411 mt 1596 411 L +1596 411 mt 1569 411 L +1569 411 mt 1542 411 L +1542 411 mt 1516 412 L +1516 412 mt 1489 412 L +1489 412 mt 1462 412 L +1462 412 mt 1435 412 L +1435 412 mt 1408 412 L +1408 412 mt 1381 412 L +1381 412 mt 1355 412 L +1355 412 mt 1328 412 L +1328 412 mt 1301 412 L +1301 412 mt 1274 412 L +1274 412 mt 1247 413 L +1247 413 mt 1221 413 L +1221 413 mt 1194 413 L +1194 413 mt 1167 413 L +1167 413 mt 1140 413 L +1140 413 mt 1113 413 L +1113 413 mt 1087 413 L +1087 413 mt 1060 413 L +1060 413 mt 1033 413 L +1033 413 mt 1006 413 L +1006 413 mt 979 413 L + 979 413 mt 953 413 L + 953 413 mt 926 413 L + 926 413 mt 899 413 L +2775 4616 mt 2775 4616 L +2775 4616 mt 2749 4614 L +2749 4614 mt 2722 4613 L +2722 4613 mt 2695 4612 L +2695 4612 mt 2668 4610 L +2668 4610 mt 2641 4609 L +2641 4609 mt 2615 4608 L +2615 4608 mt 2588 4607 L +2588 4607 mt 2561 4606 L +2561 4606 mt 2534 4606 L +2534 4606 mt 2507 4605 L +2507 4605 mt 2480 4604 L +2480 4604 mt 2454 4603 L +2454 4603 mt 2427 4603 L +2427 4603 mt 2400 4602 L +2400 4602 mt 2373 4602 L +2373 4602 mt 2346 4601 L +2346 4601 mt 2320 4601 L +2320 4601 mt 2293 4600 L +2293 4600 mt 2266 4600 L +2266 4600 mt 2239 4599 L +2239 4599 mt 2212 4599 L +2212 4599 mt 2186 4599 L +2186 4599 mt 2159 4598 L +2159 4598 mt 2132 4598 L +2132 4598 mt 2105 4598 L +2105 4598 mt 2078 4597 L +2078 4597 mt 2052 4597 L +2052 4597 mt 2025 4597 L +2025 4597 mt 1998 4596 L +1998 4596 mt 1971 4596 L +1971 4596 mt 1944 4596 L +1944 4596 mt 1918 4596 L +1918 4596 mt 1891 4596 L +1891 4596 mt 1864 4595 L +1864 4595 mt 1837 4595 L +1837 4595 mt 1810 4595 L +1810 4595 mt 1784 4595 L +1784 4595 mt 1757 4595 L +1757 4595 mt 1730 4594 L +1730 4594 mt 1703 4594 L +1703 4594 mt 1676 4594 L +1676 4594 mt 1650 4594 L +1650 4594 mt 1623 4594 L +1623 4594 mt 1596 4594 L +1596 4594 mt 1569 4594 L +1569 4594 mt 1542 4594 L +1542 4594 mt 1516 4593 L +1516 4593 mt 1489 4593 L +1489 4593 mt 1462 4593 L +1462 4593 mt 1435 4593 L +1435 4593 mt 1408 4593 L +1408 4593 mt 1381 4593 L +1381 4593 mt 1355 4593 L +1355 4593 mt 1328 4593 L +1328 4593 mt 1301 4593 L +1301 4593 mt 1274 4593 L +1274 4593 mt 1247 4592 L +1247 4592 mt 1221 4592 L +1221 4592 mt 1194 4592 L +1194 4592 mt 1167 4592 L +1167 4592 mt 1140 4592 L +1140 4592 mt 1113 4592 L +1113 4592 mt 1087 4592 L +1087 4592 mt 1060 4592 L +1060 4592 mt 1033 4592 L +1033 4592 mt 1006 4592 L +1006 4592 mt 979 4592 L + 979 4592 mt 953 4592 L + 953 4592 mt 926 4592 L + 926 4592 mt 899 4592 L +4384 389 mt 4384 389 L +4384 389 mt 4410 391 L +4410 391 mt 4437 392 L +4437 392 mt 4464 393 L +4464 393 mt 4491 395 L +4491 395 mt 4518 396 L +4518 396 mt 4544 397 L +4544 397 mt 4571 398 L +4571 398 mt 4598 399 L +4598 399 mt 4625 399 L +4625 399 mt 4652 400 L +4652 400 mt 4679 401 L +4679 401 mt 4705 402 L +4705 402 mt 4732 402 L +4732 402 mt 4759 403 L +4759 403 mt 4786 403 L +4786 403 mt 4813 404 L +4813 404 mt 4839 404 L +4839 404 mt 4866 405 L +4866 405 mt 4893 405 L +4893 405 mt 4920 406 L +4920 406 mt 4947 406 L +4947 406 mt 4973 406 L +4973 406 mt 5000 407 L +5000 407 mt 5027 407 L 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+\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{tikz} +\usetikzlibrary{shapes,positioning} +\usetikzlibrary{narrow} + +\begin{document} +\begin{tikzpicture}[auto, line width=1pt, >=narrow,x=0.9mm,y=0.9mm] + \draw (0,0) ellipse[y radius=16, x radius=32]; + \draw (0,0) ellipse[y radius=8, x radius=16]; + \draw (0,0) ellipse[y radius=4, x radius=8]; + \draw [thin,->](45:32 and 16)-- +(-80:10); + \draw [thin,->](130:32 and 16)--+(-10:15); + \draw [thin,->](-70:32 and 16)--+(170:15); + \node at (-30:30){$V$}; +\end{tikzpicture} +%Hejsan +\end{document} \ No newline at end of file diff --git a/figures/lyap_ex.tex b/figures/lyap_ex.tex new file mode 100644 index 0000000..710ed9d --- /dev/null +++ b/figures/lyap_ex.tex @@ -0,0 +1,19 @@ +\documentclass{standalone} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{tikz} +\usetikzlibrary{shapes,positioning} +\usetikzlibrary{narrow} + +\begin{document} +\begin{tikzpicture}[auto, line width=1pt, >=narrow,x=1mm,y=1mm] + \draw[thin,->](0,0)--node[yshift=3mm](){$V$}(0,12); + \draw[thin,->](0,0)--node[swap,xshift=3mm](){$x_2$}(12,0); + \draw[thin,->](0,0)--node[xshift=-2mm,yshift=-1mm](){$x_1$}(-6,-6); + \draw (20,20) arc[start angle=0, end angle=-180, radius=20mm]; + \draw(0,20) ellipse[x radius=20mm, y radius=5mm]; +\end{tikzpicture} +%Hejsan +\end{document} \ No newline at end of file diff --git a/figures/lyap_invariant.tex b/figures/lyap_invariant.tex new file mode 100644 index 0000000..54bf227 --- /dev/null +++ b/figures/lyap_invariant.tex @@ -0,0 +1,25 @@ +\documentclass[]{standalone} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{tikz} +\begin{document} +\begin{tikzpicture}[line width=1pt, x=1mm,y=1mm,scale=1.1] +% \draw[style=help lines] (-30,-10)grid (30,30); +\draw (0,0) ellipse[x radius=10mm, y radius=2mm]; +\draw[scale=0.5] (0,0) ellipse[x radius=10mm, y 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+\psfrag{x2}[][]{\hspace*{8mm}\raisebox{5mm}{$\displaystyle x_2$}} +\includegraphics[width=0.5\hsize]{pend_lyap_eps} +\end{document} \ No newline at end of file diff --git a/figures/pend_lyap_eps.eps b/figures/pend_lyap_eps.eps new file mode 100644 index 0000000..57e9f7e --- /dev/null +++ b/figures/pend_lyap_eps.eps @@ -0,0 +1,96421 @@ +%!PS-Adobe-2.0 +%%Creator: MATLAB, The Mathworks, Inc. +%%Title: /work/kalle/pend_lyap.ps +%%CreationDate: 01/19/98 11:15:31 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Pages: (atend) +%%BoundingBox: (atend) +%%EndComments + +%%BeginProlog + +% MathWorks dictionary +/MathWorks 160 dict begin + +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef + +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +/rm /rmoveto ldef +/rl /rlineto ldef +/s /show ldef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef + +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef + +% orientation switch +/portraitMode 0 def +/landscapeMode 1 def + +% coordinate system mappings +/dpi2point 0 def + +% font control +/FontSize 0 def +/FMS { + /FontSize xstore %save size off stack + findfont + [FontSize 0 0 FontSize neg 0 0] + makefont + setfont + }bdef + +/reencode { +exch dup where +{pop load} {pop StandardEncoding} ifelse +exch +dup 3 1 roll +findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def + currentdict +end +definefont pop +} bdef + +/isroman { +findfont /CharStrings get +/Agrave known +} bdef + +/FMSR { +3 1 roll 1 index +dup isroman +{reencode} {pop pop} ifelse +exch FMS +} bdef + +/csm { + 1 dpi2point div -1 dpi2point div scale + neg translate + landscapeMode eq {90 rotate} if + } bdef + +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef + +% macros for lines and objects +/L { + lineto + stroke + } bdef +/MP { + 3 1 roll moveto + 1 sub {rlineto} repeat + } bdef +/AP { + {rlineto} repeat + } bdef +/PP { + closepath eofill + } bdef +/DP { + closepath stroke + } bdef +/MR { + 4 -2 roll moveto + dup 0 exch rlineto + exch 0 rlineto + neg 0 exch rlineto + closepath + } bdef +/FR { + MR stroke + } bdef +/PR { + MR fill + } bdef +/L1i { + { currentfile picstr readhexstring pop } image + } bdef + +/tMatrix matrix def +/MakeOval { + newpath + tMatrix currentmatrix pop + translate scale + 0 0 1 0 360 arc + tMatrix setmatrix + } bdef +/FO { + MakeOval + stroke + } bdef +/PO { + MakeOval + fill + } bdef + +/PD { + currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap + } bdef + +/FA { + newpath + tMatrix currentmatrix pop + translate scale + 0 0 1 5 -2 roll arc + tMatrix setmatrix + stroke + } bdef +/PA { + newpath + tMatrix currentmatrix pop + translate 0 0 moveto scale + 0 0 1 5 -2 roll arc + closepath + tMatrix setmatrix + fill + } bdef + +/FAn { + newpath + tMatrix currentmatrix pop + translate scale + 0 0 1 5 -2 roll arcn + tMatrix setmatrix + stroke + } bdef +/PAn { + newpath + tMatrix currentmatrix pop + translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn + closepath + tMatrix setmatrix + fill + } bdef + +/MRR { + /vradius xdef + /hradius xdef + /lry xdef + /lrx xdef + /uly xdef + /ulx xdef + newpath + tMatrix currentmatrix pop + ulx hradius add uly vradius add translate + hradius vradius scale + 0 0 1 180 270 arc + tMatrix setmatrix + lrx hradius sub uly vradius add translate + hradius vradius scale + 0 0 1 270 360 arc + tMatrix setmatrix + lrx hradius sub lry vradius sub translate + hradius vradius scale + 0 0 1 0 90 arc + tMatrix setmatrix + ulx hradius add lry vradius sub translate + hradius vradius scale + 0 0 1 90 180 arc + tMatrix setmatrix + closepath + } bdef +/FRR { + MRR stroke } bdef +/PRR { + MRR fill } bdef + +/MlrRR { + /lry xdef + /lrx xdef + /uly xdef + /ulx xdef + /rad lry uly sub 2 div def + newpath + tMatrix currentmatrix pop + ulx rad add uly rad add translate + rad rad scale + 0 0 1 90 270 arc + tMatrix setmatrix + lrx rad sub lry rad sub translate + rad rad scale + 0 0 1 270 90 arc + tMatrix setmatrix + closepath + } bdef +/FlrRR { + MlrRR stroke } bdef +/PlrRR { + MlrRR fill } bdef + +/MtbRR { + /lry xdef + /lrx xdef + /uly xdef + /ulx xdef + /rad lrx ulx sub 2 div def + newpath + tMatrix currentmatrix pop + ulx rad add uly rad add translate + rad rad scale + 0 0 1 180 360 arc + tMatrix setmatrix + lrx rad sub lry rad sub translate + rad rad scale + 0 0 1 0 180 arc + tMatrix setmatrix + closepath + } bdef +/FtbRR { + MtbRR stroke } bdef +/PtbRR { + MtbRR fill } bdef + +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 62 200 563 583 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: graph1 1 +bplot + +/dpi2point 12 def +portraitMode 0216 7344 csm + + 531 341 6011 4598 MR c np +92 dict begin %Colortable dictionary +/c0 { 0 0 0 sr} bdef +/c1 { 1 1 1 sr} bdef +/c2 { 1 0 0 sr} bdef +/c3 { 0 1 0 sr} bdef +/c4 { 0 0 1 sr} bdef +/c5 { 1 1 0 sr} bdef +/c6 { 1 0 1 sr} bdef +/c7 { 0 1 1 sr} bdef +1 j +1 sg + 0 0 6920 5189 PR +6 w +0 2009 -2563 -1157 0 -2010 6260 3556 4 MP +PP +2563 1158 0 2009 -2563 -1157 0 -2010 6260 3556 5 MP stroke +2563 1158 2798 -1061 -2563 -1157 3462 4616 4 MP +PP +-2798 1060 2563 1158 2798 -1061 -2563 -1157 3462 4616 5 MP stroke +0 2009 2798 -1060 0 -2010 899 3459 4 MP +PP +-2798 1061 0 2009 2798 -1060 0 -2010 899 3459 5 MP stroke +4 w +DO +0 sg +3462 4616 mt 899 3459 L + 899 3459 mt 899 1449 L +4861 4086 mt 2298 2928 L +2298 2928 mt 2298 919 L +6260 3556 mt 3697 2398 L +3697 2398 mt 3697 389 L +3462 4616 mt 6260 3556 L +6260 3556 mt 6260 1546 L +2181 4037 mt 4978 2977 L +4978 2977 mt 4978 968 L + 899 3459 mt 3697 2398 L +3697 2398 mt 3697 389 L + 899 3459 mt 3697 2398 L +3697 2398 mt 6260 3556 L + 899 2956 mt 3697 1896 L +3697 1896 mt 6260 3053 L + 899 2454 mt 3697 1394 L +3697 1394 mt 6260 2551 L + 899 1952 mt 3697 891 L +3697 891 mt 6260 2049 L + 899 1449 mt 3697 389 L +3697 389 mt 6260 1546 L +SO +6 w +3462 4616 mt 6260 3556 L +3462 4616 mt 899 3459 L + 899 3459 mt 899 1449 L +3462 4616 mt 3531 4647 L +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 240 FMSR + +3562 4884 mt +(-2) s +4861 4086 mt 4929 4117 L +4961 4354 mt +(0) s +6260 3556 mt 6328 3586 L +6360 3824 mt +(2) s +3462 4616 mt 3393 4643 L +2953 4878 mt +(-10) s +2181 4037 mt 2111 4064 L +1945 4299 mt +(0) s + 899 3459 mt 829 3485 L + 531 3720 mt +(10) s + 899 3459 mt 831 3428 L + 666 3503 mt +(0) s + 899 2956 mt 831 2926 L + 666 3000 mt +(1) s + 899 2454 mt 831 2423 L + 666 2498 mt +(2) s + 899 1952 mt 831 1921 L + 666 1996 mt +(3) s + 899 1449 mt 831 1419 L + 666 1493 mt +(4) s +gs 899 389 5362 4228 MR c np +1 sg +-13 -31 70 -125 6190 1752 3 MP +PP +0 sg +6190 1752 mt 6260 1627 L +6260 1627 mt 6247 1596 L +1 sg +70 -125 -13 -31 6190 1752 3 MP +PP +0 sg +6190 1752 mt 6177 1721 L +6177 1721 mt 6247 1596 L +1 sg +-13 -26 70 -125 6177 1721 3 MP +PP +0 sg +6177 1721 mt 6247 1596 L +6247 1596 mt 6234 1570 L +1 sg +70 -124 -13 -27 6177 1721 3 MP +PP +0 sg +6177 1721 mt 6164 1694 L +6164 1694 mt 6234 1570 L +1 sg +-12 -22 70 -124 6164 1694 3 MP +PP +0 sg +6164 1694 mt 6234 1570 L +6234 1570 mt 6222 1548 L +1 sg +70 -124 -12 -22 6164 1694 3 MP +PP +0 sg +6164 1694 mt 6152 1672 L +6152 1672 mt 6222 1548 L +1 sg +-13 -17 70 -124 6152 1672 3 MP +PP +0 sg +6152 1672 mt 6222 1548 L +6222 1548 mt 6209 1531 L +1 sg +70 -124 -13 -17 6152 1672 3 MP +PP +0 sg +6152 1672 mt 6139 1655 L +6139 1655 mt 6209 1531 L +1 sg +-13 -12 70 -124 6139 1655 3 MP +PP +0 sg +6139 1655 mt 6209 1531 L +6209 1531 mt 6196 1519 L +1 sg +70 -124 -13 -12 6139 1655 3 MP +PP +0 sg +6139 1655 mt 6126 1643 L +6126 1643 mt 6196 1519 L +1 sg +-13 -7 70 -124 6126 1643 3 MP +PP +0 sg +6126 1643 mt 6196 1519 L +6196 1519 mt 6183 1512 L +1 sg +-13 -31 70 -119 6120 1871 3 MP +PP +0 sg +6120 1871 mt 6190 1752 L +6190 1752 mt 6177 1721 L +1 sg +70 -124 -13 -7 6126 1643 3 MP +PP +0 sg +6126 1643 mt 6113 1636 L +6113 1636 mt 6183 1512 L +1 sg +-13 -2 70 -124 6113 1636 3 MP +PP +0 sg +6113 1636 mt 6183 1512 L +6183 1512 mt 6170 1510 L +1 sg +-13 -27 70 -119 6107 1840 3 MP +PP +0 sg +6107 1840 mt 6177 1721 L +6177 1721 mt 6164 1694 L +1 sg +70 -119 -13 -31 6120 1871 3 MP +PP +0 sg +6120 1871 mt 6107 1840 L +6107 1840 mt 6177 1721 L +1 sg +70 -124 -13 -2 6113 1636 3 MP +PP +0 sg +6113 1636 mt 6100 1634 L +6100 1634 mt 6170 1510 L +1 sg +-13 3 70 -124 6100 1634 3 MP +PP +0 sg +6100 1634 mt 6170 1510 L +6170 1510 mt 6157 1513 L +1 sg +-12 -22 70 -120 6094 1814 3 MP +PP +0 sg +6094 1814 mt 6164 1694 L +6164 1694 mt 6152 1672 L +1 sg +70 -120 -13 -26 6107 1840 3 MP +PP +0 sg +6107 1840 mt 6094 1814 L +6094 1814 mt 6164 1694 L +1 sg +-12 8 69 -124 6088 1637 3 MP +PP +0 sg +6088 1637 mt 6157 1513 L +6157 1513 mt 6145 1521 L +1 sg +69 -124 -12 3 6100 1634 3 MP +PP +0 sg +6100 1634 mt 6088 1637 L +6088 1637 mt 6157 1513 L +1 sg +-13 -17 70 -120 6082 1792 3 MP +PP +0 sg +6082 1792 mt 6152 1672 L +6152 1672 mt 6139 1655 L +1 sg +70 -120 -12 -22 6094 1814 3 MP +PP +0 sg +6094 1814 mt 6082 1792 L +6082 1792 mt 6152 1672 L +1 sg +70 -124 -13 8 6088 1637 3 MP +PP +0 sg +6088 1637 mt 6075 1645 L +6075 1645 mt 6145 1521 L +1 sg +-13 12 70 -124 6075 1645 3 MP +PP +0 sg +6075 1645 mt 6145 1521 L +6145 1521 mt 6132 1533 L +1 sg +70 -120 -13 -17 6082 1792 3 MP +PP +0 sg +6082 1792 mt 6069 1775 L +6069 1775 mt 6139 1655 L +1 sg +-13 -12 70 -120 6069 1775 3 MP +PP +0 sg +6069 1775 mt 6139 1655 L +6139 1655 mt 6126 1643 L +1 sg +-12 30 70 -125 6023 1722 3 MP +PP +0 sg +6023 1722 mt 6093 1597 L +6093 1597 mt 6081 1627 L +1 sg +-13 17 70 -125 6062 1658 3 MP +PP +0 sg +6062 1658 mt 6132 1533 L +6132 1533 mt 6119 1550 L +1 sg +-13 32 70 -124 6011 1751 3 MP +PP +0 sg +6011 1751 mt 6081 1627 L +6081 1627 mt 6068 1659 L +1 sg +70 -124 -12 29 6023 1722 3 MP +PP +0 sg +6023 1722 mt 6011 1751 L +6011 1751 mt 6081 1627 L +1 sg +70 -125 -13 13 6075 1645 3 MP +PP +0 sg +6075 1645 mt 6062 1658 L +6062 1658 mt 6132 1533 L +1 sg +70 -120 -13 -12 6069 1775 3 MP +PP +0 sg +6069 1775 mt 6056 1763 L +6056 1763 mt 6126 1643 L +1 sg +-13 -7 70 -120 6056 1763 3 MP +PP +0 sg +6056 1763 mt 6126 1643 L +6126 1643 mt 6113 1636 L +1 sg +-13 -31 70 -114 6050 1985 3 MP +PP +0 sg +6050 1985 mt 6120 1871 L +6120 1871 mt 6107 1840 L +1 sg +70 -125 -13 26 6036 1696 3 MP +PP +0 sg +6036 1696 mt 6023 1722 L 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1734 mt 6029 1774 L +1 sg +70 -120 -13 -2 6043 1756 3 MP +PP +0 sg +6043 1756 mt 6030 1754 L +6030 1754 mt 6100 1634 L +1 sg +-13 39 70 -125 5985 1820 3 MP +PP +0 sg +5985 1820 mt 6055 1695 L +6055 1695 mt 6042 1734 L +1 sg +-12 3 70 -120 6030 1754 3 MP +PP +0 sg +6030 1754 mt 6100 1634 L +6100 1634 mt 6088 1637 L +1 sg +69 -114 -12 -27 6037 1955 3 MP +PP +0 sg +6037 1955 mt 6025 1928 L +6025 1928 mt 6094 1814 L +1 sg +-12 -22 69 -114 6025 1928 3 MP +PP +0 sg +6025 1928 mt 6094 1814 L +6094 1814 mt 6082 1792 L +1 sg +70 -125 -13 36 5998 1784 3 MP +PP +0 sg +5998 1784 mt 5985 1820 L +5985 1820 mt 6055 1695 L +1 sg +70 -125 -13 33 6011 1751 3 MP +PP +0 sg +6011 1751 mt 5998 1784 L +5998 1784 mt 6068 1659 L +1 sg +-13 36 70 -125 5998 1784 3 MP +PP +0 sg +5998 1784 mt 6068 1659 L +6068 1659 mt 6055 1695 L +1 sg +-13 8 70 -120 6018 1757 3 MP +PP +0 sg +6018 1757 mt 6088 1637 L +6088 1637 mt 6075 1645 L +1 sg +70 -120 -12 3 6030 1754 3 MP +PP +0 sg +6030 1754 mt 6018 1757 L +6018 1757 mt 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L +3204 3555 mt 3274 3551 L +1 sg +-13 16 70 1 3224 3543 3 MP +PP +0 sg +3224 3543 mt 3294 3544 L +3294 3544 mt 3281 3560 L +1 sg +-13 -43 70 46 3235 3512 3 MP +PP +0 sg +3235 3512 mt 3305 3558 L +3305 3558 mt 3292 3515 L +1 sg +70 57 -13 29 3301 3484 3 MP +PP +0 sg +3301 3484 mt 3288 3513 L +3288 3513 mt 3358 3570 L +1 sg +-13 25 70 57 3288 3513 3 MP +PP +0 sg +3288 3513 mt 3358 3570 L +3358 3570 mt 3345 3595 L +1 sg +70 -84 -13 -52 3298 1764 3 MP +PP +0 sg +3298 1764 mt 3285 1712 L +3285 1712 mt 3355 1628 L +1 sg +-13 -50 70 -84 3285 1712 3 MP +PP +0 sg +3285 1712 mt 3355 1628 L +3355 1628 mt 3342 1578 L +1 sg +70 -69 -13 -41 3242 2606 3 MP +PP +0 sg +3242 2606 mt 3229 2565 L +3229 2565 mt 3299 2496 L +1 sg +70 -49 -13 44 3270 2536 3 MP +PP +0 sg +3270 2536 mt 3257 2580 L +3257 2580 mt 3327 2531 L +1 sg +-13 44 70 -49 3270 2536 3 MP +PP +0 sg +3270 2536 mt 3340 2487 L +3340 2487 mt 3327 2531 L +1 sg +70 -29 -13 -45 3298 2582 3 MP +PP +0 sg +3298 2582 mt 3285 2537 L +3285 2537 mt 3355 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2895 L +2296 2895 mt 2366 2966 L +1 sg +70 6 -13 1 2308 3208 3 MP +PP +0 sg +2308 3208 mt 2295 3209 L +2295 3209 mt 2365 3215 L +1 sg +70 -9 -13 -52 2249 2426 3 MP +PP +0 sg +2249 2426 mt 2236 2374 L +2236 2374 mt 2306 2365 L +1 sg +-13 27 70 31 2253 2344 3 MP +PP +0 sg +2253 2344 mt 2323 2375 L +2323 2375 mt 2310 2402 L +1 sg +-13 -38 70 42 2293 2374 3 MP +PP +0 sg +2293 2374 mt 2363 2416 L +2363 2416 mt 2350 2378 L +1 sg +-13 31 70 31 2240 2371 3 MP +PP +0 sg +2240 2371 mt 2310 2402 L +2310 2402 mt 2297 2433 L +1 sg +70 31 -13 27 2253 2344 3 MP +PP +0 sg +2253 2344 mt 2240 2371 L +2240 2371 mt 2310 2402 L +1 sg +70 42 -13 -42 2306 2416 3 MP +PP +0 sg +2306 2416 mt 2293 2374 L +2293 2374 mt 2363 2416 L +1 sg +70 -24 -13 10 2266 1984 3 MP +PP +0 sg +2266 1984 mt 2253 1994 L +2253 1994 mt 2323 1970 L +1 sg +70 -24 -13 5 2279 1979 3 MP +PP +0 sg +2279 1979 mt 2266 1984 L +2266 1984 mt 2336 1960 L +1 sg +-13 10 70 -24 2266 1984 3 MP +PP +0 sg +2266 1984 mt 2336 1960 L +2336 1960 mt 2323 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-13 -41 1988 2229 3 MP +PP +0 sg +1988 2229 mt 1975 2188 L +1975 2188 mt 2045 2194 L +1 sg +70 51 -13 23 1986 2146 3 MP +PP +0 sg +1986 2146 mt 1973 2169 L +1973 2169 mt 2043 2220 L +1 sg +-13 23 70 51 1986 2146 3 MP +PP +0 sg +1986 2146 mt 2056 2197 L +2056 2197 mt 2043 2220 L +1 sg +-13 -42 70 61 2026 2201 3 MP +PP +0 sg +2026 2201 mt 2096 2262 L +2096 2262 mt 2083 2220 L +1 sg +-12 27 70 51 1973 2169 3 MP +PP +0 sg +1973 2169 mt 2043 2220 L +2043 2220 mt 2031 2247 L +1 sg +70 61 -13 -44 2039 2245 3 MP +PP +0 sg +2039 2245 mt 2026 2201 L +2026 2201 mt 2096 2262 L +1 sg +-12 10 69 -4 1987 2030 3 MP +PP +0 sg +1987 2030 mt 2056 2026 L +2056 2026 mt 2044 2036 L +1 sg +70 -3 -13 -10 2038 2038 3 MP +PP +0 sg +2038 2038 mt 2025 2028 L +2025 2028 mt 2095 2025 L +1 sg +69 -4 -12 6 1999 2024 3 MP +PP +0 sg +1999 2024 mt 1987 2030 L +1987 2030 mt 2056 2026 L +1 sg +-13 5 70 -3 1999 2024 3 MP +PP +0 sg +1999 2024 mt 2069 2021 L +2069 2021 mt 2056 2026 L +1 sg +-13 -5 70 -3 2025 2028 3 MP +PP +0 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L +1980 2043 mt 2050 2099 L +1 sg +-13 -6 70 57 1993 2048 3 MP +PP +0 sg +1993 2048 mt 2063 2105 L +2063 2105 mt 2050 2099 L +1 sg +-13 -16 70 57 2019 2074 3 MP +PP +0 sg +2019 2074 mt 2089 2131 L +2089 2131 mt 2076 2115 L +1 sg +70 57 -13 -20 2032 2094 3 MP +PP +0 sg +2032 2094 mt 2019 2074 L +2019 2074 mt 2089 2131 L +1 sg +-12 -5 69 27 2016 3134 3 MP +PP +0 sg +2016 3134 mt 2085 3161 L +2085 3161 mt 2073 3156 L +1 sg +69 27 -12 1 2028 3133 3 MP +PP +0 sg +2028 3133 mt 2016 3134 L +2016 3134 mt 2085 3161 L +1 sg +-12 -56 70 16 1937 2560 3 MP +PP +0 sg +1937 2560 mt 2007 2576 L +2007 2576 mt 1995 2520 L +1 sg +70 16 -13 -56 1950 2616 3 MP +PP +0 sg +1950 2616 mt 1937 2560 L +1937 2560 mt 2007 2576 L +1 sg +70 41 -13 45 2011 2561 3 MP +PP +0 sg +2011 2561 mt 1998 2606 L +1998 2606 mt 2068 2647 L +1 sg +-13 -55 70 71 2014 2590 3 MP +PP +0 sg +2014 2590 mt 2084 2661 L +2084 2661 mt 2071 2606 L +1 sg +70 71 -13 -53 2027 2643 3 MP +PP +0 sg +2027 2643 mt 2014 2590 L +2014 2590 mt 2084 2661 L +1 sg +70 51 -13 19 1999 2127 3 MP +PP +0 sg +1999 2127 mt 1986 2146 L +1986 2146 mt 2056 2197 L +1 sg +-13 -37 70 61 2013 2159 3 MP +PP +0 sg +2013 2159 mt 2083 2220 L +2083 2220 mt 2070 2183 L +1 sg +70 61 -13 -42 2026 2201 3 MP +PP +0 sg +2026 2201 mt 2013 2159 L +2013 2159 mt 2083 2220 L +1 sg +-13 1 70 -4 2012 2024 3 MP +PP +0 sg +2012 2024 mt 2082 2020 L +2082 2020 mt 2069 2021 L +1 sg +70 -4 -13 -4 2025 2028 3 MP +PP +0 sg +2025 2028 mt 2012 2024 L +2012 2024 mt 2082 2020 L +1 sg +70 31 -12 24 2022 3027 3 MP +PP +0 sg +2022 3027 mt 2010 3051 L +2010 3051 mt 2080 3082 L +1 sg +69 21 -12 -28 2021 3090 3 MP +PP +0 sg +2021 3090 mt 2009 3062 L +2009 3062 mt 2078 3083 L +1 sg +-13 20 70 31 2010 3051 3 MP +PP +0 sg +2010 3051 mt 2080 3082 L +2080 3082 mt 2067 3102 L +1 sg +70 66 -13 -54 2020 2384 3 MP +PP +0 sg +2020 2384 mt 2007 2330 L +2007 2330 mt 2077 2396 L +1 sg +-13 -52 70 66 2007 2330 3 MP +PP +0 sg +2007 2330 mt 2077 2396 L +2077 2396 mt 2064 2344 L +1 sg +70 56 -13 -15 2019 2074 3 MP +PP +0 sg +2019 2074 mt 2006 2059 L +2006 2059 mt 2076 2115 L +1 sg +-13 -10 70 56 2006 2059 3 MP +PP +0 sg +2006 2059 mt 2076 2115 L +2076 2115 mt 2063 2105 L +1 sg +-12 -50 70 21 1944 2861 3 MP +PP +0 sg +1944 2861 mt 2014 2882 L +2014 2882 mt 2002 2832 L +1 sg +-13 37 70 37 2004 2859 3 MP +PP +0 sg +2004 2859 mt 2074 2896 L +2074 2896 mt 2061 2933 L +1 sg +70 37 -13 39 2017 2820 3 MP +PP +0 sg +2017 2820 mt 2004 2859 L +2004 2859 mt 2074 2896 L +1 sg +70 26 -13 -4 2016 3134 3 MP +PP +0 sg +2016 3134 mt 2003 3130 L +2003 3130 mt 2073 3156 L +1 sg +70 16 -12 -56 1937 2560 3 MP +PP +0 sg +1937 2560 mt 1925 2504 L +1925 2504 mt 1995 2520 L +1 sg +-13 45 70 46 1941 2515 3 MP +PP +0 sg +1941 2515 mt 2011 2561 L +2011 2561 mt 1998 2606 L +1 sg +70 16 -13 -55 1925 2504 3 MP +PP +0 sg +1925 2504 mt 1912 2449 L +1912 2449 mt 1982 2465 L +1 sg +-13 -55 70 16 1925 2504 3 MP +PP +0 sg +1925 2504 mt 1995 2520 L +1995 2520 mt 1982 2465 L +1 sg +70 46 -13 44 1954 2471 3 MP +PP +0 sg +1954 2471 mt 1941 2515 L +1941 2515 mt 2011 2561 L +1 sg +-13 44 70 46 1954 2471 3 MP +PP +0 sg +1954 2471 mt 2024 2517 L +2024 2517 mt 2011 2561 L +1 sg +70 71 -13 -55 2014 2590 3 MP +PP +0 sg +2014 2590 mt 2001 2535 L +2001 2535 mt 2071 2606 L +1 sg +-13 -55 70 71 2001 2535 3 MP +PP +0 sg +2001 2535 mt 2071 2606 L +2071 2606 mt 2058 2551 L +1 sg +70 1 -13 19 1891 2054 3 MP +PP +0 sg +1891 2054 mt 1878 2073 L +1878 2073 mt 1948 2074 L +1 sg +69 6 -12 -24 1936 2089 3 MP +PP +0 sg +1936 2089 mt 1924 2065 L +1924 2065 mt 1993 2071 L +1 sg +70 6 -13 -29 1949 2118 3 MP +PP +0 sg +1949 2118 mt 1936 2089 L +1936 2089 mt 2006 2095 L +1 sg +-13 -24 70 6 1936 2089 3 MP +PP +0 sg +1936 2089 mt 2006 2095 L +2006 2095 mt 1993 2071 L +1 sg +-13 14 70 56 1942 2057 3 MP +PP +0 sg +1942 2057 mt 2012 2113 L +2012 2113 mt 1999 2127 L +1 sg +-13 -34 70 62 2000 2121 3 MP +PP +0 sg +2000 2121 mt 2070 2183 L +2070 2183 mt 2057 2149 L +1 sg +70 62 -13 -38 2013 2159 3 MP +PP +0 sg +2013 2159 mt 2000 2121 L +2000 2121 mt 2070 2183 L +1 sg +70 -3 -13 0 2012 2024 3 MP +PP +0 sg +2012 2024 mt 1999 2024 L +1999 2024 mt 2069 2021 L +1 sg +-12 -28 70 26 1951 3064 3 MP +PP +0 sg +1951 3064 mt 2021 3090 L +2021 3090 mt 2009 3062 L +1 sg +70 26 -13 -24 1964 3088 3 MP +PP +0 sg +1964 3088 mt 1951 3064 L +1951 3064 mt 2021 3090 L +1 sg +-13 -24 70 26 1964 3088 3 MP +PP +0 sg +1964 3088 mt 2034 3114 L +2034 3114 mt 2021 3090 L +1 sg +-13 15 70 32 1997 3070 3 MP +PP +0 sg +1997 3070 mt 2067 3102 L +2067 3102 mt 2054 3117 L +1 sg +70 26 -12 -28 1951 3064 3 MP +PP +0 sg +1951 3064 mt 1939 3036 L +1939 3036 mt 2009 3062 L +1 sg +-13 -33 70 26 1939 3036 3 MP +PP +0 sg +1939 3036 mt 2009 3062 L +2009 3062 mt 1996 3029 L +1 sg +70 32 -13 19 2010 3051 3 MP +PP +0 sg +2010 3051 mt 1997 3070 L +1997 3070 mt 2067 3102 L +1 sg +70 11 -13 -44 1931 2262 3 MP +PP +0 sg +1931 2262 mt 1918 2218 L +1918 2218 mt 1988 2229 L +1 sg +69 11 -12 -47 1943 2309 3 MP +PP +0 sg +1943 2309 mt 1931 2262 L +1931 2262 mt 2000 2273 L +1 sg +-12 -44 69 11 1931 2262 3 MP +PP +0 sg +1931 2262 mt 2000 2273 L +2000 2273 mt 1988 2229 L +1 sg +-13 33 70 52 1948 2226 3 MP +PP +0 sg +1948 2226 mt 2018 2278 L +2018 2278 mt 2005 2311 L +1 sg +-13 -51 70 67 1994 2277 3 MP +PP +0 sg +1994 2277 mt 2064 2344 L +2064 2344 mt 2051 2293 L +1 sg +-13 37 70 51 1935 2260 3 MP +PP +0 sg +1935 2260 mt 2005 2311 L +2005 2311 mt 1992 2348 L +1 sg +70 51 -13 34 1948 2226 3 MP +PP +0 sg +1948 2226 mt 1935 2260 L +1935 2260 mt 2005 2311 L +1 sg +70 67 -13 -53 2007 2330 3 MP +PP +0 sg +2007 2330 mt 1994 2277 L +1994 2277 mt 2064 2344 L +1 sg +70 57 -13 -11 2006 2059 3 MP +PP +0 sg +2006 2059 mt 1993 2048 L +1993 2048 mt 2063 2105 L +1 sg +70 21 -13 -44 1970 2952 3 MP +PP +0 sg +1970 2952 mt 1957 2908 L +1957 2908 mt 2027 2929 L +1 sg +-13 -47 70 21 1957 2908 3 MP +PP +0 sg +1957 2908 mt 2027 2929 L +2027 2929 mt 2014 2882 L +1 sg +-13 -44 70 21 1970 2952 3 MP +PP +0 sg +1970 2952 mt 2040 2973 L +2040 2973 mt 2027 2929 L +1 sg +-13 35 70 36 1991 2897 3 MP +PP +0 sg +1991 2897 mt 2061 2933 L +2061 2933 mt 2048 2968 L +1 sg +70 21 -13 -47 1957 2908 3 MP +PP +0 sg +1957 2908 mt 1944 2861 L +1944 2861 mt 2014 2882 L +1 sg +70 36 -13 38 2004 2859 3 MP +PP +0 sg +2004 2859 mt 1991 2897 L +1991 2897 mt 2061 2933 L +1 sg +-13 -54 70 16 1912 2449 3 MP +PP +0 sg +1912 2449 mt 1982 2465 L +1982 2465 mt 1969 2411 L +1 sg +70 46 -12 43 1966 2428 3 MP +PP +0 sg +1966 2428 mt 1954 2471 L +1954 2471 mt 2024 2517 L +1 sg +-12 42 70 47 1966 2428 3 MP +PP +0 sg +1966 2428 mt 2036 2475 L +2036 2475 mt 2024 2517 L +1 sg +-12 -56 70 72 1988 2479 3 MP +PP +0 sg +1988 2479 mt 2058 2551 L +2058 2551 mt 2046 2495 L +1 sg +70 72 -13 -56 2001 2535 3 MP +PP +0 sg +2001 2535 mt 1988 2479 L +1988 2479 mt 2058 2551 L +1 sg +70 56 -13 10 1955 2047 3 MP +PP +0 sg +1955 2047 mt 1942 2057 L +1942 2057 mt 2012 2113 L +1 sg +-13 9 70 57 1955 2047 3 MP +PP +0 sg +1955 2047 mt 2025 2104 L +2025 2104 mt 2012 2113 L +1 sg +-13 -30 70 62 1987 2087 3 MP +PP +0 sg +1987 2087 mt 2057 2149 L +2057 2149 mt 2044 2119 L +1 sg +70 62 -13 -34 2000 2121 3 MP +PP +0 sg +2000 2121 mt 1987 2087 L +1987 2087 mt 2057 2149 L +1 sg +70 26 -13 -19 1977 3107 3 MP +PP +0 sg +1977 3107 mt 1964 3088 L +1964 3088 mt 2034 3114 L +1 sg +-13 11 70 31 1984 3086 3 MP +PP +0 sg +1984 3086 mt 2054 3117 L +2054 3117 mt 2041 3128 L +1 sg +70 31 -13 16 1997 3070 3 MP +PP +0 sg +1997 3070 mt 1984 3086 L +1984 3086 mt 2054 3117 L +1 sg +-13 -41 70 11 1918 2218 3 MP +PP +0 sg +1918 2218 mt 1988 2229 L +1988 2229 mt 1975 2188 L +1 sg +70 52 -13 30 1961 2196 3 MP +PP +0 sg +1961 2196 mt 1948 2226 L +1948 2226 mt 2018 2278 L +1 sg +70 11 -13 -41 1918 2218 3 MP +PP +0 sg +1918 2218 mt 1905 2177 L +1905 2177 mt 1975 2188 L +1 sg +-13 -37 70 11 1905 2177 3 MP +PP +0 sg +1905 2177 mt 1975 2188 L +1975 2188 mt 1962 2151 L +1 sg +70 51 -12 27 1973 2169 3 MP +PP +0 sg +1973 2169 mt 1961 2196 L +1961 2196 mt 2031 2247 L +1 sg +-13 31 70 51 1961 2196 3 MP +PP +0 sg +1961 2196 mt 2031 2247 L +2031 2247 mt 2018 2278 L +1 sg +-12 -48 69 66 1982 2227 3 MP +PP +0 sg +1982 2227 mt 2051 2293 L +2051 2293 mt 2039 2245 L +1 sg +69 66 -12 -50 1994 2277 3 MP +PP +0 sg +1994 2277 mt 1982 2227 L +1982 2227 mt 2051 2293 L +1 sg +70 37 -13 34 1991 2897 3 MP +PP +0 sg +1991 2897 mt 1978 2931 L +1978 2931 mt 2048 2968 L +1 sg +70 21 -13 -40 1983 2992 3 MP +PP +0 sg +1983 2992 mt 1970 2952 L +1970 2952 mt 2040 2973 L +1 sg +-13 31 70 37 1978 2931 3 MP +PP +0 sg +1978 2931 mt 2048 2968 L +2048 2968 mt 2035 2999 L +1 sg +70 47 -13 41 1979 2387 3 MP +PP +0 sg +1979 2387 mt 1966 2428 L +1966 2428 mt 2036 2475 L +1 sg +70 72 -12 -56 1988 2479 3 MP +PP +0 sg +1988 2479 mt 1976 2423 L +1976 2423 mt 2046 2495 L +1 sg +-13 -56 70 72 1976 2423 3 MP +PP +0 sg +1976 2423 mt 2046 2495 L +2046 2495 mt 2033 2439 L +1 sg +70 57 -13 4 1968 2043 3 MP +PP +0 sg +1968 2043 mt 1955 2047 L +1955 2047 mt 2025 2104 L +1 sg +-12 -25 69 61 1975 2058 3 MP +PP +0 sg +1975 2058 mt 2044 2119 L +2044 2119 mt 2032 2094 L +1 sg +69 61 -12 -29 1987 2087 3 MP +PP +0 sg +1987 2087 mt 1975 2058 L +1975 2058 mt 2044 2119 L +1 sg +70 41 -13 44 1985 2650 3 MP +PP +0 sg +1985 2650 mt 1972 2694 L +1972 2694 mt 2042 2735 L +1 sg +-13 43 70 41 1972 2694 3 MP +PP +0 sg +1972 2694 mt 2042 2735 L +2042 2735 mt 2029 2778 L +1 sg +70 32 -13 10 1984 3086 3 MP +PP +0 sg +1984 3086 mt 1971 3096 L +1971 3096 mt 2041 3128 L +1 sg +-13 5 70 32 1971 3096 3 MP +PP +0 sg +1971 3096 mt 2041 3128 L +2041 3128 mt 2028 3133 L +1 sg +70 66 -13 -48 1982 2227 3 MP +PP +0 sg +1982 2227 mt 1969 2179 L +1969 2179 mt 2039 2245 L +1 sg +-13 -44 70 66 1969 2179 3 MP +PP +0 sg +1969 2179 mt 2039 2245 L +2039 2245 mt 2026 2201 L +1 sg +-13 -10 70 1 1968 2037 3 MP +PP +0 sg +1968 2037 mt 2038 2038 L +2038 2038 mt 2025 2028 L +1 sg +70 26 -13 -40 1913 2966 3 MP +PP +0 sg +1913 2966 mt 1900 2926 L +1900 2926 mt 1970 2952 L +1 sg +-13 -44 70 26 1900 2926 3 MP +PP +0 sg +1900 2926 mt 1970 2952 L +1970 2952 mt 1957 2908 L +1 sg +-13 -40 70 26 1913 2966 3 MP +PP +0 sg +1913 2966 mt 1983 2992 L +1983 2992 mt 1970 2952 L +1 sg +70 36 -13 32 1978 2931 3 MP +PP +0 sg +1978 2931 mt 1965 2963 L +1965 2963 mt 2035 2999 L +1 sg +-13 28 70 36 1965 2963 3 MP +PP +0 sg +1965 2963 mt 2035 2999 L +2035 2999 mt 2022 3027 L +1 sg +70 16 -13 -50 1886 2343 3 MP +PP +0 sg +1886 2343 mt 1873 2293 L +1873 2293 mt 1943 2309 L +1 sg +-12 -47 70 16 1873 2293 3 MP +PP +0 sg +1873 2293 mt 1943 2309 L +1943 2309 mt 1931 2262 L +1 sg +-13 -50 70 16 1886 2343 3 MP +PP +0 sg +1886 2343 mt 1956 2359 L +1956 2359 mt 1943 2309 L +1 sg +70 16 -13 -52 1899 2395 3 MP +PP +0 sg +1899 2395 mt 1886 2343 L +1886 2343 mt 1956 2359 L +1 sg +-13 -52 70 16 1899 2395 3 MP +PP +0 sg +1899 2395 mt 1969 2411 L +1969 2411 mt 1956 2359 L +1 sg +70 51 -13 39 1922 2297 3 MP +PP +0 sg +1922 2297 mt 1909 2336 L +1909 2336 mt 1979 2387 L +1 sg +-13 39 70 51 1922 2297 3 MP +PP +0 sg +1922 2297 mt 1992 2348 L +1992 2348 mt 1979 2387 L +1 sg +-13 -55 70 72 1963 2367 3 MP +PP +0 sg +1963 2367 mt 2033 2439 L +2033 2439 mt 2020 2384 L +1 sg +70 16 -13 -54 1912 2449 3 MP +PP +0 sg +1912 2449 mt 1899 2395 L +1899 2395 mt 1969 2411 L +1 sg +-13 41 70 51 1909 2336 3 MP +PP +0 sg +1909 2336 mt 1979 2387 L +1979 2387 mt 1966 2428 L +1 sg +70 72 -13 -56 1976 2423 3 MP +PP +0 sg +1976 2423 mt 1963 2367 L +1963 2367 mt 2033 2439 L +1 sg +-13 10 70 6 1847 2023 3 MP +PP +0 sg +1847 2023 mt 1917 2029 L +1917 2029 mt 1904 2039 L +1 sg +70 1 -12 6 1929 2023 3 MP +PP +0 sg +1929 2023 mt 1917 2029 L +1917 2029 mt 1987 2030 L +1 sg +-13 -10 70 6 1898 2031 3 MP +PP +0 sg +1898 2031 mt 1968 2037 L +1968 2037 mt 1955 2027 L +1 sg +70 6 -13 -14 1911 2045 3 MP +PP +0 sg +1911 2045 mt 1898 2031 L +1898 2031 mt 1968 2037 L +1 sg +-12 6 70 6 1859 2017 3 MP +PP +0 sg +1859 2017 mt 1929 2023 L +1929 2023 mt 1917 2029 L +1 sg +70 6 -13 0 1872 2017 3 MP +PP +0 sg +1872 2017 mt 1859 2017 L +1859 2017 mt 1929 2023 L +1 sg +-13 0 70 6 1872 2017 3 MP +PP +0 sg +1872 2017 mt 1942 2023 L +1942 2023 mt 1929 2023 L +1 sg +-12 6 70 1 1929 2023 3 MP +PP +0 sg +1929 2023 mt 1999 2024 L +1999 2024 mt 1987 2030 L +1 sg +70 1 -13 0 1942 2023 3 MP +PP +0 sg +1942 2023 mt 1929 2023 L +1929 2023 mt 1999 2024 L +1 sg +-13 -4 70 6 1885 2021 3 MP +PP +0 sg +1885 2021 mt 1955 2027 L +1955 2027 mt 1942 2023 L +1 sg +70 6 -12 6 1859 2017 3 MP +PP +0 sg +1859 2017 mt 1847 2023 L +1847 2023 mt 1917 2029 L +1 sg +70 6 -13 -10 1898 2031 3 MP +PP +0 sg +1898 2031 mt 1885 2021 L +1885 2021 mt 1955 2027 L +1 sg +-13 0 70 1 1942 2023 3 MP +PP +0 sg +1942 2023 mt 2012 2024 L +2012 2024 mt 1999 2024 L +1 sg +70 1 -13 -4 1955 2027 3 MP +PP +0 sg +1955 2027 mt 1942 2023 L +1942 2023 mt 2012 2024 L +1 sg +-13 -4 70 1 1955 2027 3 MP +PP +0 sg +1955 2027 mt 2025 2028 L +2025 2028 mt 2012 2024 L +1 sg +70 1 -13 -10 1968 2037 3 MP +PP +0 sg +1968 2037 mt 1955 2027 L +1955 2027 mt 2025 2028 L +1 sg +-12 0 70 61 1910 1982 3 MP +PP +0 sg +1910 1982 mt 1980 2043 L +1980 2043 mt 1968 2043 L +1 sg +-13 -20 70 61 1962 2033 3 MP +PP +0 sg +1962 2033 mt 2032 2094 L +2032 2094 mt 2019 2074 L +1 sg +70 62 -12 -1 1910 1982 3 MP +PP +0 sg +1910 1982 mt 1898 1981 L +1898 1981 mt 1968 2043 L +1 sg +70 6 -13 -4 1885 2021 3 MP +PP +0 sg +1885 2021 mt 1872 2017 L +1872 2017 mt 1942 2023 L +1 sg +-13 4 70 62 1898 1981 3 MP +PP +0 sg +1898 1981 mt 1968 2043 L +1968 2043 mt 1955 2047 L +1 sg +70 61 -13 -25 1975 2058 3 MP +PP +0 sg +1975 2058 mt 1962 2033 L +1962 2033 mt 2032 2094 L +1 sg +-13 -54 70 21 1919 2759 3 MP +PP +0 sg +1919 2759 mt 1989 2780 L +1989 2780 mt 1976 2726 L +1 sg +-12 42 70 41 1959 2737 3 MP +PP +0 sg +1959 2737 mt 2029 2778 L +2029 2778 mt 2017 2820 L +1 sg +70 21 -13 -54 1919 2759 3 MP +PP +0 sg +1919 2759 mt 1906 2705 L +1906 2705 mt 1976 2726 L +1 sg +-13 -55 70 21 1906 2705 3 MP +PP +0 sg +1906 2705 mt 1976 2726 L +1976 2726 mt 1963 2671 L +1 sg +70 41 -13 43 1972 2694 3 MP +PP +0 sg +1972 2694 mt 1959 2737 L +1959 2737 mt 2029 2778 L +1 sg +-13 -19 70 31 1907 3076 3 MP +PP +0 sg +1907 3076 mt 1977 3107 L +1977 3107 mt 1964 3088 L +1 sg +70 31 -13 -14 1920 3090 3 MP +PP +0 sg +1920 3090 mt 1907 3076 L +1907 3076 mt 1977 3107 L +1 sg +-13 -14 70 31 1920 3090 3 MP +PP +0 sg +1920 3090 mt 1990 3121 L +1990 3121 mt 1977 3107 L +1 sg +70 31 -13 6 1971 3096 3 MP +PP +0 sg +1971 3096 mt 1958 3102 L +1958 3102 mt 2028 3133 L +1 sg +-12 1 70 31 1958 3102 3 MP +PP +0 sg +1958 3102 mt 2028 3133 L +2028 3133 mt 2016 3134 L +1 sg +-13 -33 70 11 1892 2140 3 MP +PP +0 sg +1892 2140 mt 1962 2151 L +1962 2151 mt 1949 2118 L +1 sg +70 11 -13 -37 1905 2177 3 MP +PP +0 sg +1905 2177 mt 1892 2140 L +1892 2140 mt 1962 2151 L +1 sg +70 11 -13 -28 1879 2106 3 MP +PP +0 sg +1879 2106 mt 1866 2078 L +1866 2078 mt 1936 2089 L +1 sg +-13 -29 70 12 1879 2106 3 MP +PP +0 sg +1879 2106 mt 1949 2118 L +1949 2118 mt 1936 2089 L +1 sg +70 12 -13 -34 1892 2140 3 MP +PP +0 sg +1892 2140 mt 1879 2106 L +1879 2106 mt 1949 2118 L +1 sg +70 56 -13 23 1916 2090 3 MP +PP +0 sg +1916 2090 mt 1903 2113 L +1903 2113 mt 1973 2169 L +1 sg +-12 27 70 56 1903 2113 3 MP +PP +0 sg +1903 2113 mt 1973 2169 L +1973 2169 mt 1961 2196 L +1 sg +-13 23 70 56 1916 2090 3 MP +PP +0 sg +1916 2090 mt 1986 2146 L +1986 2146 mt 1973 2169 L +1 sg +70 67 -13 -45 1969 2179 3 MP +PP +0 sg +1969 2179 mt 1956 2134 L +1956 2134 mt 2026 2201 L +1 sg +-13 -42 70 67 1956 2134 3 MP +PP +0 sg +1956 2134 mt 2026 2201 L +2026 2201 mt 2013 2159 L +1 sg +70 26 -13 -37 1926 3003 3 MP +PP +0 sg +1926 3003 mt 1913 2966 L +1913 2966 mt 1983 2992 L +1 sg +-13 -37 70 26 1926 3003 3 MP +PP +0 sg +1926 3003 mt 1996 3029 L +1996 3029 mt 1983 2992 L +1 sg +69 36 -12 28 1965 2963 3 MP +PP +0 sg +1965 2963 mt 1953 2991 L +1953 2991 mt 2022 3027 L +1 sg +70 26 -13 -33 1939 3036 3 MP +PP +0 sg +1939 3036 mt 1926 3003 L +1926 3003 mt 1996 3029 L +1 sg +-12 24 69 36 1953 2991 3 MP +PP +0 sg +1953 2991 mt 2022 3027 L +2022 3027 mt 2010 3051 L +1 sg +70 51 -13 37 1935 2260 3 MP +PP +0 sg +1935 2260 mt 1922 2297 L +1922 2297 mt 1992 2348 L +1 sg +-13 -54 70 72 1950 2312 3 MP +PP +0 sg +1950 2312 mt 2020 2384 L +2020 2384 mt 2007 2330 L +1 sg +70 72 -13 -55 1963 2367 3 MP +PP +0 sg +1963 2367 mt 1950 2312 L +1950 2312 mt 2020 2384 L +1 sg +70 61 -13 -5 1923 1987 3 MP +PP +0 sg +1923 1987 mt 1910 1982 L +1910 1982 mt 1980 2043 L +1 sg +-13 -5 70 61 1923 1987 3 MP +PP +0 sg +1923 1987 mt 1993 2048 L +1993 2048 mt 1980 2043 L +1 sg +70 61 -13 -20 1962 2033 3 MP +PP +0 sg +1962 2033 mt 1949 2013 L +1949 2013 mt 2019 2074 L +1 sg +-13 -15 70 61 1949 2013 3 MP +PP +0 sg +1949 2013 mt 2019 2074 L +2019 2074 mt 2006 2059 L +1 sg +70 21 -13 -52 1932 2811 3 MP +PP +0 sg +1932 2811 mt 1919 2759 L +1919 2759 mt 1989 2780 L +1 sg +-13 -52 70 21 1932 2811 3 MP +PP +0 sg +1932 2811 mt 2002 2832 L +2002 2832 mt 1989 2780 L +1 sg +70 21 -12 -50 1944 2861 3 MP +PP +0 sg +1944 2861 mt 1932 2811 L +1932 2811 mt 2002 2832 L +1 sg +-13 39 70 42 1947 2778 3 MP +PP +0 sg +1947 2778 mt 2017 2820 L +2017 2820 mt 2004 2859 L +1 sg +70 42 -12 41 1959 2737 3 MP +PP +0 sg +1959 2737 mt 1947 2778 L +1947 2778 mt 2017 2820 L +1 sg +70 31 -13 -9 1933 3099 3 MP +PP +0 sg +1933 3099 mt 1920 3090 L +1920 3090 mt 1990 3121 L +1 sg +-13 -9 70 31 1933 3099 3 MP +PP +0 sg +1933 3099 mt 2003 3130 L +2003 3130 mt 1990 3121 L +1 sg +-13 -4 70 31 1946 3103 3 MP +PP +0 sg +1946 3103 mt 2016 3134 L +2016 3134 mt 2003 3130 L +1 sg +70 31 -12 1 1958 3102 3 MP +PP +0 sg +1958 3102 mt 1946 3103 L +1946 3103 mt 2016 3134 L +1 sg +-12 -24 70 11 1866 2078 3 MP +PP +0 sg +1866 2078 mt 1936 2089 L +1936 2089 mt 1924 2065 L +1 sg +70 56 -13 19 1929 2071 3 MP +PP +0 sg +1929 2071 mt 1916 2090 L +1916 2090 mt 1986 2146 L +1 sg +-13 19 70 56 1929 2071 3 MP +PP +0 sg +1929 2071 mt 1999 2127 L +1999 2127 mt 1986 2146 L +1 sg +70 66 -13 -41 1956 2134 3 MP +PP +0 sg +1956 2134 mt 1943 2093 L +1943 2093 mt 2013 2159 L +1 sg +70 56 -13 14 1942 2057 3 MP +PP +0 sg +1942 2057 mt 1929 2071 L +1929 2071 mt 1999 2127 L +1 sg +-13 -38 70 66 1943 2093 3 MP +PP +0 sg +1943 2093 mt 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