From 0b6b837b15b57c47b616e8c8dd7c93d78449f318 Mon Sep 17 00:00:00 2001 From: AnnetteSchmidt Date: Mon, 6 Apr 2020 19:02:34 +0200 Subject: [PATCH] fist commit --- .gitignore | 276 ++++++++ Abbildungen/SOK_TEG_FS.pdf | Bin 0 -> 489462 bytes ENT4_FS.pdf | Bin 0 -> 348529 bytes ENT4_FS.tex | 1259 ++++++++++++++++++++++++++++++++++++ README.md | 3 + 5 files changed, 1538 insertions(+) create mode 100644 .gitignore create mode 100644 Abbildungen/SOK_TEG_FS.pdf create mode 100644 ENT4_FS.pdf create mode 100644 ENT4_FS.tex create mode 100644 README.md diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..238bc81 --- /dev/null +++ b/.gitignore @@ -0,0 +1,276 @@ +## 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 + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Comment the next line if you want to keep your tikz graphics 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 + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# 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 \ No newline at end of file diff --git a/Abbildungen/SOK_TEG_FS.pdf b/Abbildungen/SOK_TEG_FS.pdf new file mode 100644 index 0000000000000000000000000000000000000000..27d107a27215aa4e4b66ee981fe3421bf7d74a55 GIT binary patch literal 489462 zcmZsC1CT6BxAoYzjWcI#+qP}nwr$%!W7}tJ+qU)3{qBqR-XHNgqPj9`=gz%zt-Z3V zJE}?LghZ(6X;@%M_x7jfVWH^h3FruH4J=@}xM-!!ZH)CD%sq^a21zIU%8&fAU z0(vI4f6{UJcaXv}y{*CbX)W^aP9q^gru&baF7( zw}yo>+cGdP(nHYGGtkpJ0*!~^AxNl$7h3BAil;*7CrKn%gH1DH2QaAxYuNT1t0hQi z0v2*;@{|4rm=G136Yxj_9(mw5iseb?$&wZY63^wY*Nm9<02Gg8+t;4qN5Y7Y9yZAW ze9;ULm%huQmz+Vs(1b;TG!97O`>AZA2lQr)rbY_9A4mSpNSD3^4nYTC8;3p%3-v?J ze!W=)pc?)=bLgzQ|VJ`3wB$dbb5B~=w{~`Im#2{_#q;I6}q)+fK z@v_cNRzLD^Bw+bZ9mwgM{NklwP^(%Z0!VW-8324XxQlJ=n0tVY3P`L zLNGD0)6g*xFw!&7u(8r>(@OuSNdKRKbTmKn{2VMl7CHh}_Mdlc+8^2dr14*AD;PW4 zIy)E||CbJJ9Te^K4UPX1r6L^x(|;2E5u&i0lc=K8j|zVbqW>8FUB9RzJpt2y(*H-S zKg;{ak3pA?fcal3{j`6Q{09Psf37nSF#qfK(`F=K{uj5OHWLBMe**n8hM9o%UtcAq zpM?KKT}g?afc4)IN=gg_tp7pkC!(B#t)Zf^lP2xYZbqwQ?B?_%zaO3clPAG{iSr+Z z{udknYYX~6YF5PD!O@9;ftB&U>z0|7nf_<9V`Tc5UReVRMJMb3UB3sJt5c*S&c(Wi z0N%v>d6b4R05E_!B zqCH9OQe8X?zeEiWg^?#y@B2pZeO~$oZ1KE(#)LiZ-_Lo!_5;5kJH~r>z3wK7nR(iZ zz;nBMXn8*m96ko(@x7idmt*lh&T6N9?@#YTr}4Sn-kiC<4{0B-z;n4?F6T{kdA#nt z3yXZ&bnp*k;TMmSIf6 zWATJXo*mBMeLfF%(YLi8-XDIX>~-@{d+aHl`J%(_BmfTXyRd&&+T-(bxKFsoula+CayAhyMGLy%k?$?uIqh!Fnj2Bb9yhT`*m)76Gt$; 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+\documentclass[7pt,a4paper,landscape]{article} + \usepackage[left=0.55cm,right=0.55cm,top=1.10cm,bottom=0.55cm,landscape, + headsep=2mm]{geometry} + + \usepackage[dvipsnames]{xcolor} %fürs farbige markieren mit \colorbox + \usepackage{lastpage} + \usepackage{fancyhdr} + \usepackage{multicol} + \usepackage[utf8]{inputenc} + \usepackage[ngerman]{babel} + \usepackage[T1]{fontenc} + \usepackage{listings} + \usepackage{enumitem} + \setitemize{leftmargin=15pt} + \setenumerate{leftmargin=15pt} + \usepackage{titlesec} + \usepackage{color,soul} + \usepackage{graphicx} + \usepackage{tabularx} + \usepackage{tikz} + \usetikzlibrary{automata,positioning,calc,matrix} %calc und Matrix für 3x3 det + \usepackage[babel,german=quotes]{csquotes} + \usepackage{arydshln} + \usepackage[fleqn]{amsmath} + \usepackage{setspace} + \usepackage{amssymb} + \usepackage{float} + \usepackage{booktabs} + \usepackage{multirow} + \usepackage{pbox} + \usepackage{pifont} + \usepackage{wrapfig} + \usepackage[T1]{fontenc} + \renewcommand*\familydefault{\sfdefault} %% Only if the base font of the document is to be sans serif + \usepackage{comment} + \usepackage{mathrsfs} %für geschwungendes Laplace L + \usepackage{trfsigns} %für Laplace-Trafo Symbol + \usepackage[hyphens]{url} + \usepackage{gensymb} %fürs degree Zeichen + \usepackage{mdframed} %colored frames + \usepackage[ + pdftex, + pdftitle={AUT5A_FS}, + pdfauthor={Annette Schmidt}, + pdfcreator={Annette Schmidt}, + pdfsubject={Formelsammlung AUT5A Regelungstechnik}, + %pdfkeywords={MPC, FCS-MPC, field weakening}, + linktoc=all, % Sowohl Text als auch Seitenzahlen als Link + colorlinks=false, % Keine Farbe bei Links + pdfborder={0 0 0}, % Kein Rand um Links + breaklinks, % Links umbrechen + bookmarks, % Lesezeichen beim Öffnen des Dokuments anzeigen + plainpages=false, % Zur korrekten Erstellung der Bookmarks + pdfpagelabels, % Zur korrekten Erstellung der Bookmarks + hypertexnames=false % Zur korrekten Erstellung der Bookmarks + ]{hyperref} + + % Header + \pagestyle{fancy} + \fancyhead{} + \fancyfoot{} + \fancyhead[L]{FS ENT4 SoSe2020, Annette Schmidt, \url{https://git.efi.th-nuernberg.de/gitea/schmidtan65529/AUT5AFormelsammlung.git} Fehler bitte sofort melden!} + \fancyhead[R]{Seite $\thepage$ von $\pageref{LastPage}$} + \fancyheadoffset{0cm} + + % Document + \setlength{\columnseprule}{0.5pt} + \setlength{\topskip}{10pt} + \setlist{nosep} + + \titleformat*{\section}{\normalsize\bfseries} + \titleformat*{\subsection}{\small\bfseries} + \titleformat*{\subsubsection}{\small\bfseries} + \titleformat*{\paragraph}{\bfseries} + \titleformat*{\subparagraph}{\bfseries} + + \titlespacing*{\section} + {0pt}{4pt}{0pt} + \titlespacing*{\subsection} + {0pt}{4pt}{0pt} + \titlespacing*{\subsubsection} + {0pt}{4pt}{0pt} + \titlespacing*{\paragraph} + {0pt}{4pt}{8pt} + + \newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}} + \newcolumntype{M}[1]{>{\centering\arraybackslash}m{#1}} + + \makeatletter + \newcommand{\xRightarrow}[2][]{\ext@arrow 0359\Rightarrowfill@{#1}{#2}} + \makeatother + + % Building blocks + %\newcommand{\heading}[1]{\noindent\section*{\colorbox{SpringGreen}{\framebox[\columnwidth][l]{#1}}}} + \newcommand{\heading}[1]{\noindent\section*{ + \fcolorbox{black}{SpringGreen}{ + \parbox{0.945\columnwidth}{#1} + }}} + + \newcommand{\subheading}[1]{\noindent\subsection*{\fcolorbox{black}{SpringGreen!50}{ + \parbox{0.945\columnwidth}{#1} + }}} + %\newcommand{\heading}[1]{\noindent\section*{\colorbox{SpringGreen}{\framebox[\columnwidth][l]{#1}}}} + %\newcommand{\subheading}[1]{\noindent\subsection*{\framebox[\columnwidth][l]{#1}}} + + \newcommand{\subsubheading}[1]{\noindent\framebox[\columnwidth][l]{#1}} + % centering stuff + \newcommand{\ccontent}[1]{\parbox{\columnwidth}{\centering{#1}}} + % for partial derivative at a point + \newcommand*{\at}[2][]{#1|_{#2}} + %for yellow highlights in equations + \newcommand{\highlight}[1]{\colorbox{yellow}{$\displaystyle #1$}} + + \graphicspath{{Abbildungen/}} %Fügt den Pfad der Abbildungen hinzu + + % Content +\begin{document} +\footnotesize +\begin{multicols*}{3} + + \heading{Grundlagen} + \begin{tabular}{ r | l } + linear & in dB \\ + \hline + $V_{lin}$ & $V_{dB} = 20 log_{10}V_{lin}$ \\ + $V_{lin} = 10^{V_{dB}/20dB}$ & $V_{dB}$ \\ + % \hline + % 1 & 0 dB\\ + % 10 & 20 dB\\ + % 2 & 6 dB\\ + % 0,1 & -20 dB\\ + % 0,5 & -6 dB\\ + % $\sqrt{2} = 2^{0,5}$ & 6 dB $\cdot 0,5 = 3$ dB\\ + \end{tabular}\\ + + \subsubheading{Einschleifiger Regelkreis} + Übertragunsfunktion allgemein: $\frac{Produkt~der~Vorw"artsglieder}{1\pm Produkt~der~Schleifenglieder}$ + bei Gegenkopplung (Minus in Schleife) - bei Mitkopplung + + \subsubheading{Zustandsregelung} + \textbf{Zustandsgleichung}\\ + $\underbrace{\left[ \begin{array}{r} \dot{x}_1(t) \\\dot{x}_2(t) \\\dot{x}_3(t) \\\end{array}\right]}_{\dot{x}(t)} = \underbrace{\left[ \begin{array}{rrr} a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33} \\\end{array}\right]}_{\textbf{A}} \cdot \underbrace{\left[ \begin{array}{r} x_1(t) \\x_2(t) \\x_3(t) \\\end{array}\right]}_{x(t)} + \underbrace{\left[ \begin{array}{r} b_1 \\b_2 \\b_3 \\\end{array}\right]}_{\textbf{B}}\cdot y(t)$\\ + \textbf{Ausgangsgleichung}\\ + $v(t) = \underbrace{[c_1~c_2~c_3]}_{\textbf{C}} \cdot x(t) + D_0\cdot y(t)$\\ + \textbf{Bestimmung der Übertragunsfunktion}\\ + Zeitkontinuierlich\\ + $G(s) = \frac{Ausgang(s)}{Eingang(s)} = \frac{V(s)}{Y(s)} = \textbf{C}(s\cdot \textbf{I} - \textbf{A})^{-1}\textbf{B} + D = \frac{\textbf{C}\cdot \text{adj}(s\cdot I - \textbf{A})\cdot\textbf{B}}{\text{det}(s\cdot I - A)} + d$\\ + Zeitdiskret\\ + $G(z) = \frac{\textbf{C}\cdot \text{adj}(z\cdot I - \textbf{A})\cdot\textbf{B}}{\text{det}(z\cdot I - A)} + D$\\ + \textbf{A} = Wirkung der Zustände aufeinander, \textbf{B} = Wirkung des Eingangssignals auf die Zustände, \textbf{C} = Wirkung der Zustände auf den Ausgang, \textbf{D} Wirkung des Eingangs auf den Ausgang (meist 0), $\dot{\textbf{x}}$ = Änderung eines Zustands, \textbf{x} = Zustände, $y$ = Stellsignals, $v$ = Ausgang, $w$ = Eingangssignal\\ + \textbf{Rechenregeln}\\ + $M^{-1} = \frac{\text{adj}(M)}{\text{det}(M)};~~$ $\text{det}= \left( \begin{array}{rr} a & b \\c & d \\\end{array}\right) = ad - cb$\\ + $\text{adj}\left( \begin{array}{rr} a & b \\c & d \\\end{array}\right) = \left( \begin{array}{rr} d & -b \\-c & a \\\end{array}\right)$\\ + $(a~b)\cdot\left( \begin{array}{rr} c & d \\e & f \\\end{array}\right) = (ac+be~ad+bf)$\\ + + $\left(\begin{array}{rr} c & d \\e & f \\\end{array}\right) \cdot \left(\begin{array}{r} a \\b \\\end{array}\right) = \left(\begin{array}{r} ca+db \\ae+bf \\\end{array}\right)$\\ + + $\left(\begin{array}{rr} a & b \\c & d \\\end{array}\right) \cdot \left(\begin{array}{rr} e & f \\g & h \\\end{array}\right) = \left(\begin{array}{rr} ae+bg & af+bh \\ce+dg & cf+dh \\\end{array}\right)$ + + $M^T$ ist die transponierte Matrix, hier wird die Matrix an der Hauptdiagonalen gespiegelt. + + \begin{flalign*} + det + \left(\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\ + a_{21} & a_{22} & a_{23} \\ + a_{31} & a_{32} & a_{33} \\ \\ + \end{array}\right) &= + a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32} &\\ + &- a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12} & + \end{flalign*} + \begin{tikzpicture} + \matrix [% + matrix of math nodes, + column sep=1em, + row sep=1em + ] (sarrus) {% + a_{11} & a_{12} & a_{13} & a_{11} & a_{12} \\ + a_{21} & a_{22} & a_{23} & a_{21} & a_{22} \\ + a_{31} & a_{32} & a_{33} & a_{31} & a_{32} \\ + }; + + \path ($(sarrus-1-3.north east)+(0.5em,0)$) edge[dotted] ($(sarrus-3-3.south east)+(0.5em,0)$) + (sarrus-1-1) edge (sarrus-2-2) + (sarrus-2-2) edge (sarrus-3-3) + (sarrus-1-2) edge (sarrus-2-3) + (sarrus-2-3) edge (sarrus-3-4) + (sarrus-1-3) edge (sarrus-2-4) + (sarrus-2-4) edge (sarrus-3-5) + (sarrus-3-1) edge[dashed] (sarrus-2-2) + (sarrus-2-2) edge[dashed] (sarrus-1-3) + (sarrus-3-2) edge[dashed] (sarrus-2-3) + (sarrus-2-3) edge[dashed] (sarrus-1-4) + (sarrus-3-3) edge[dashed] (sarrus-2-4) + (sarrus-2-4) edge[dashed] (sarrus-1-5); + + \foreach \c in {1,2,3} {\node[anchor=south] at (sarrus-1-\c.north) {$+$};}; + \foreach \c in {1,2,3} {\node[anchor=north] at (sarrus-3-\c.south) {$-$};}; + \end{tikzpicture} + + \subsubheading{Linearisieren} + Betriebspunkt (BP) Linearisierung:\\ + Definition Kleinsignal: $\underbrace{\Delta y(t)}_{Kleinsignal} = \underbrace{y(t)}_{Großsignal} - \underbrace{y_{BP}}_{BP} \equiv$ Abweichung vom Betriebspunkt\\ + + Beispiel DGLs:\\ + \begin{equation}\label{eq:bsp1} + \dot{v}(t)= \underbrace{-\alpha_Z (t) a}_{linear} + \underbrace{M(t)b }_{linear} \underbrace{- c}_{nicht~linear} - \underbrace{d\cdot v^2(t)}_{nicht~linear} = f_1 + \end{equation} + \begin{equation}\label{eq:bsp2} + \dot{M}(t) = \underbrace{-\frac{1}{T}M(t) + \frac{M_{max}}{T}y(t)}_{linear} = f_2 + \end{equation} + Betriebspunkt finden:\\ + Eingangsgrößen müssen gegeben sein, dann die zeitlichen Ableitungen der DGL gleich Null gesetzt, da im Betriebspunkt keine Änderung stattfindet und die BPs der Eingangsgrößen einsetzen. Also im Beispiel $\dot{v}(BP) = 0$ und $\dot{M}(BP) = 0$. Mit $z(t) = 0$. Natürlich vor der eigentlichen Linearisierung.\\ + + Linearisierung:\\ + aus \eqref{eq:bsp1}: + \begin{flalign*} + \Delta \dot{v}(t) &= \frac{\partial f_1}{\partial \alpha_Z} \bigg|_{BP} \Delta \alpha_Z(t) + \frac{\partial f_1}{\partial M} \bigg|_{BP} \Delta M(t) + \frac{\partial f_1}{\partial v} \bigg|_{BP} \Delta v(t) &\\ + &= -a \Delta \alpha_Z(t) + b \Delta M(t) - 2d v_{BP} \Delta v(t) & + \end{flalign*} + aus \eqref{eq:bsp2}: + \begin{flalign*} + \Delta \dot{M}(t) &= \frac{\partial f_2}{\partial M} \bigg|_{BP} \Delta M(t) + \frac{\partial f_2}{\partial y} \bigg|_{BP} \Delta y(t) &\\ + &= -\frac{1}{T} \Delta M(t) + \frac{M_{max}}{T} \Delta y(t) & + \end{flalign*} + Ableitungsregeln: Siehe Seite 9\\ + \subsubheading{Laplace Transformation} + \ccontent{\includegraphics[height=5cm]{LaplaceTrafo2.png}} + \ccontent{\includegraphics[height=13.9cm]{LaplaceTrafo.png}} + $a\cdot t ~~\laplace~~ \frac{a}{s^2}$\\ + \subsubheading{Bleibende Regelabweichung /Schleppfehler} + \textbf{Führungsverhalten:} Sollwert = 1, Störwert = 0\\ + Bei Eingangssprung: $w(t) = \sigma(t)$ also $W(s) = \frac{1}{s}$ + \begin{flalign*} + \underset{t\rightarrow \infty}{lim} e(t) &= \underset{t\rightarrow \infty}{lim} ~(w(t) - v(t)) &\\ + &= \underset{s\rightarrow 0}{lim}~ s(W(s) - V(s)) = \underset{s\rightarrow 0}{lim} ~s(W(s)- G_W(s)W(s)) & + \end{flalign*} + \textbf{Störverhalten:} Sollwert = 0, Störwert = 1\\ + Sollwert = $w(t) = 0$, $V(s) = \frac{1}{s} G_z(s)$ + \begin{flalign*} + \underset{t\rightarrow \infty}{lim} e(t) &= \underset{t\rightarrow \infty}{lim} ~(w(t) - v(t)) = \underset{t\rightarrow \infty}{lim} ~( - v(t)) &\\ + &= -\underset{s\rightarrow 0}{lim}~ s V(s) = -\underset{s\rightarrow 0}{lim} ~s \frac{1}{s} G_z(s) = -G_z(0) & + \end{flalign*} + \textbf{Achtung! Immer alles ausmultiplizieren!} + \newpage + \heading{Zustandsregler} + \subheading{Grundgleichungen} + \subsubheading{1. Schritt: Zustandsdarstellung eines Systems} + Übertragungsfunktion, Eigenwerte\\ + \textbf{Zustandsdifferentialgleichung}\\ + \colorbox{yellow}{$\dot{\textbf{x}}(t) = \textbf{Ax}(t) + \textbf{B}y(t)$} (linear und zeitinvariant)\\ + \textbf{Ausgansgleichung}\\ + $v(t) = \textbf{Cx}(t) + \textbf{D}y(t)$\\ + Mit \textbf{Systemmatrix A, Eingangsvektor B, Ausgansmatrix C, Durchgriff D} wobei meistens \textbf{D} = 0. \\ + Durch Laplace-Trafo: + \begin{flalign*} + s\textbf{X}(s) &= \textbf{AX}(s) + \textbf{B}Y(s) &\\ + s\textbf{X}(s) - \textbf{AX}(s) &= + \textbf{B}Y(s)~~~~~~|\cdot \textbf{I}&\\ + s \textbf{I} \cdot \textbf{X}(s) - \textbf{AX}(s) &= + \textbf{B}Y(s)\\ + s(\textbf{I} - \textbf{A})^{-1}\textbf{X}(s) &= \textbf{B}Y(s) &\\ + \rightarrow \textbf{X}(s) &= (s\textbf{I} - \textbf{A})^{-1} \textbf{B}Y(s) + \end{flalign*} + % $s\textbf{X}(s) = \textbf{AX}(s) + \textbf{B}Y(s)$\\ + % $s\textbf{X}(s) - \textbf{AX}(s) = + \textbf{B}Y(s)~~~~~~|\cdot \textbf{I}$\\ + % $s \textbf{I} \cdot \textbf{X}(s) - \textbf{AX}(s) = + \textbf{B}Y(s)$\\ + % $s(\textbf{I} - \textbf{A})^{-1}\textbf{X}(s) = \textbf{B}Y(s)$\\ + % $\rightarrow \textbf{X}(s) = (s\textbf{I} - \textbf{A})^{-1} \textbf{B}Y(s)$\\ + Mit der Ausgansgleichung erhält man die Übertragungsfunktion von $Y(s)$ nach $V(s)$: + \begin{flalign*} + V(s) &= \textbf{CX}(s) + \textbf{D}Y(s) = \underbrace{(\textbf{C}(s\textbf{I}- \textbf{A})^{-1}\textbf{B} + \textbf{D}}_{G_s(s)})Y(s) &\\ + &= G_s(s)Y(s) & + \end{flalign*} + Bei Inversion ($\textbf{M}^{-1}$) wird durch die Determinante der Matrix dividiert. Deswegen können die Pole der Übertragungsfunktion nur bei den Eigenwerten der Systemmatrix \textbf{A} liegen: $det(s\textbf{I}- \textbf{A}) = 0$ + + \subsubheading{2. Schritt: Zustandsdarstellung eines P-geregelten Systems} + Eigenwerte\\ + Stellsignal eines mit P-Regler geregelten Systems:\\ + $y(t) = K_P (w(t) - v(t)) ~\laplace ~ Y(s) = K_P(W(s)-V(s)) = K_P(W(s)- \textbf{C}X(s))$\\ + in der Zustandsgleichung resultiert: + \begin{flalign*} + s\textbf{X}(s) &= \textbf{AX}(s) + \textbf{B}Y(s) &\\ + &= \underbrace{(\textbf{A} - \textbf{B}K_P\textbf{C})}_{\textbf{A}_{RK}}\textbf{X}(s) + \underbrace{(\textbf{B}K_P)}_{\textbf{B}_{RK}}W(s) & + \end{flalign*} + mit Systemmatrix $\textbf{A}_{RK}$ und Eingangsvektor $\textbf{B}_{RK}$ des Regelkreises. Mit Ausgangsgleichung der Regelstrecke $V(s) = \textbf{CX}(s)$ (bei $\textbf{D} = 0$) erhält man die Führungsübertragungsfunktion des Regelkreises: + \begin{flalign*} + V(s) &= \underbrace{(\textbf{C}(s\textbf{I} - \textbf{A} + \textbf{B}K_P\textbf{C})^{-1}\textbf{B}K_P)}_{G_W(s)}Y(s) &\\ + &= G_W(s)Y(s) & + \end{flalign*} + Jetzt sind die Eigenwerte des Regelkreises (und damit die Pole von $G_W(s)$) gleich: \\ + $det(s\textbf{I} - \textbf{A} + \textbf{B}K_P \textbf{C}) = det(s\textbf{I} - \textbf{A}_{RK}) = 0$\\ + Dies ist die charakteristische Gleichung des Regelkreises mit P-Regler. Beschreibt man die Strecke und den Regler mit Übertragungsfunktionen, so ist die charakteristische Gleichung $1 + G_R(s) G_S(s) = 0$ äquivalten zur vorherigen. + + \subsubheading{3. Schritt: Vollständige Zustandsrückführung} + für Regelstrecke in der 2. kanonischen Form = Regelungsnormalform der Regelstrecke\\ + Strecke ohne Regler: $G(s) = \frac{\mathcal{L}\{v(t)\}}{\mathcal{L}\{y(t)\}} = \frac{V(s)}{Y(s)} = \frac{b_2s^2 + b_1s +b_0}{s^n + a_2s^2 + a_1s + a_0}$\\ mit $n=3$ weil 3. Ordnung.\\ + Zustandsgleichung: + \begin{equation*} + \dot{x}(t) = + \underbrace{ + \left[ + \begin{array}{ccc} + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + -a_0 & -a_1 & -a_2 \\ + \end{array} + \right]}_{\textbf{A}_{RNF}} + \cdot x(t) + + \underbrace{ + \left[ + \begin{array}{c} + 0 \\ + 0 \\ + 1 \\ + \end{array} + \right]}_{\textbf{B}_{RNF}} + \cdot y(t) + \end{equation*} + Ausgangsgleichung: + \begin{equation*} + v(t) =\left[\begin{array}{rrr} b_0&b_1&b_2 \end{array}\right] x(t) + \end{equation*} + Zustandsrückführung:\\ + $y(t) = -k_1x_1(t) - k_2x_2(t) - k_3x_3(t) + \ell w(t) = - \left[k_1~k_2~k_3\right] \textbf{x}(t) + \ell w(t) = -\textbf{Kx}(t) + \ell w(t)$\\ + \ccontent{\includegraphics[height=5cm]{ZustangsRegler3Ordnung.png}} + Zustandsgleichung mit Zustandsregler: + \begin{flalign*} + \dot{x}(t) &= + \underbrace{ + \left[ + \begin{array}{ccc} + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + -(a_0 + k_1) & -(a_1+k_2) & -(a_2 + k_3) \\ + \end{array} + \right]}_{\textbf{A}_{RK}} + x(t) & \\ &+ + \underbrace{ + \left[ + \begin{array}{c} + 0 \\ + 0 \\ + 1 \\ + \end{array} + \right]\ell}_{\textbf{B}_{RK}} + w(t) & + \end{flalign*} + Ausgansgleichung unverändert: $v(t) =\left[\begin{array}{rrr} b_0&b_1&b_2 \end{array}\right] x(t)$. Somit ergibt sich die Führungsübertragungsfunktion $G_w(s) = \frac{\mathcal{L}\{v(t)\}}{\mathcal{L}\{y(t)\}} = \frac{V(s)}{Y(s)} = \frac{\left(b_2s^2 + b_1s +b_0\right) \ell}{s^n + \left(a_2 + k_3\right)s^2 + \left(a_1 + k_2\right)s + \left(a_0 + k_1\right)} = \underbrace{\highlight{\ell \frac{Z_s(s)}{\tilde{N}(s)}}}_{Vorgabepolynom}$ \\ + Streckennullstellen sind auch Nullstellen von $G_w(s)$ und durch geeignete Wahl von $k_1...k_n$ kann ein beliebiges Nennerpolynom, d.h. eine beliebige Polverteilung realisiert werden.\\ + Man sieht: + \begin{enumerate} + \item Bei einer Zustandsregelung mit Strecke in RNF erscheint das Zählerpolynom der ungeregelten Strecke wieder in der Führungsübertragungsfunktion, hier jedoch mit dem Faktor $\ell$ mulitpliziert. + \item Das heißt: die Lage der Nullstellen wird durch die Zustandsrückführung nicht verändert. + \item Im Nennerpolynom $n$-ten Grades sind die $n$ Rückführkoeffizienten $k_i$ zu finden, die die Lage aller $n$ Pole beeinflussen. + \end{enumerate} + Wenn die Regelstrecke $G(s)$ und damit $a_0$ bis $a_n$ und die Pole des ZR gegeben sind, kann über Koeffizientenvergleich ($\tilde{N}(s) = s-s_\infty ... = s^n + \left(a_2 + k_3\right)s^2 + \left(a_1 + k_2\right)s + \left(a_0 + k_1\right)$) $k_n$ bestimmt werden. $\ell$ wird in der Regel so gewählt, dass die Regelung sationär genau ist also $G_w(s\rightarrow 0) = 1$ ist. + + \subsubheading{4. Schritt ZR in beliebiger Zustandsdarstellung} + Strecke: $\dot{\textbf{x}}(t) = \textbf{Ax}(t) + \textbf{B}y(t)$ mit $y(t) = -\textbf{Kx}(t) + \ell w(t)$ ergibt $\dot{\textbf{x}}(t) = \textbf{Ax}(t) - \textbf{BKx}(t) + \textbf{B}\ell w(t) = \left(\textbf{A} - \textbf{BK} \right)\textbf{x}(t) + \textbf{B}\ell w(t)$\\ + Entwurfsgleichung: \\ + \colorbox{orange!60}{$\underbrace{det(s\textbf{I} - \textbf{A} + \textbf{B}\textbf{K})}_{charak. Polynom des ZR-Kreises} \overset{!}{=} \underbrace{\tilde{N}(s)}_{Vorgabepolynom}$}\\ + es gilt die Führungsübertragungsfunktion:\\ + \colorbox{yellow}{$G_w(s) = \textbf{C}(s\textbf{I} - \textbf{A} + \textbf{BK})^{-1} \textbf{B}\ell = \ell \frac{Z(s)}{\tilde{N}(s)}$}\\ + \colorbox{yellow}{Bestimmung von $\ell$ aus $G_w(s = 0) \overset{!}{=} 1$}\\ + $\rightarrow$ stationär genau, wennn $G_w(s = 0) \overset{!}{=} 1$\\ + \colorbox{yellow}{$\ell = \frac{1}{\textbf{C}(-\textbf{A} + \textbf{BK})^{-1}\textbf{B}}$ bzw. $\ell = \frac{\tilde{N}(0)}{Z_s(0)}$} + \subheading{Steuerbarkeit} + %\textbf{Definition: Steuerbarkeit}\\ + Ein System heißt vollständig steuerbar, wenn jeder Anfangszustand $x(t_0)$ in endlicher Zeit $t_1 > t_0$ durch ein unbeschränktes Stellsignal $y(t)$ in jeden beliebigen Endzustand $x(t_1)$ überführt werden kann.\\ + Man kann zeigen, dass diese Bedingung erfüllt ist, wenn die Steuerbarkeitsmatrix\\ + $\textbf{Q}_s = \left[\textbf{B}~~\textbf{AB}~~\textbf{A}^2\textbf{B}~...~\textbf{A}^{n-1}\textbf{B}\right]$\\ + vollen Rang hat, d.h. invertierbar ist, d.h. ihre Determinante ist ungleich null. + \subheading{Beobachter} + Wenn nicht alle Zustandgrößen messbar sind wird ein Beobachter gebraucht. \\ + \ccontent{\includegraphics[height=5cm]{BeobachterFarbe2.png}} + Im ersten Schritt wird dafür ein \textcolor{red}{Parallelmodell} der Regelstrecke erstellt. Da dieses in den meisten Fällen nicht exakt ist, wird im zweiten Schritt noch eine \textcolor{violet}{Korrektur} der Modellzustände aud Messungen benötigt. \\ + \textbf{Differentialgleichung für Beobachter-Fehler:}\\ + $\textbf{e}(t) = \textbf{x}(t) - \hat{\textbf{x}}(t)$\\ + $\dot{\textbf{x}}(t) = \textbf{Ax}(t) + \textbf{B}y(t) + \underbrace{\textbf{G}z(t)}_{St"orung}$\\ + $\hat{\dot{\textbf{x}}}(t) = \textbf{A}\hat{\textbf{x}}(t) + \textbf{B}y(t) + \textbf{H}(v(t) - \hat{v}(t))$\\ + $\dot{\textbf{e}}(t) = \underbrace{(\textbf{A}- \textbf{HC})}_{\textbf{A}_{Beobachter}} \textbf{e}(t) + \textbf{G}z(t)$\\ + $\textbf{A}_R = \textbf{A} - \textbf{BK}$ und $\textbf{A}_B = \textbf{A} - \textbf{HC}$\\ + \begin{itemize} + \item $y(t)$ regt einen Beobachtungsfehler an + \item Beobachtungsfehler klingen ab mit der Dynamik $\textbf{A}_B = \textbf{A} - \textbf{HC}$ + \item Nicht messbare Störungen auf der Regelstrecken regen Beobachtungsfehler an! + \item Durch \textbf{H} lässt sich eine beliebige Dynamik für das Abklingen von Anfangswertfehlern einstellen. + \end{itemize} + Entwurfsgleichung: Festlegung der Parameter des Rückführvektors \textbf{H} über vorgegebene Beobachterpole\\ + $det(s\textbf{I} - \textbf{A} + \textbf{HC}) = \underset{n}{\overset{i = 1}{\prod}} (s-s_{B,i})= \Delta(s)$ \\ + Das Vorgabepolynom $\Delta (s)$ charakterisiert die Beobachterdynamik, genauso wie oben das Polynom $\tilde{N}(s)$ die Regelungsdynamik charakterisiert. + + \subheading{Beobachtbarkeit und Regelbarkeit} + \subsubheading{Beobachtbarkeit} + Ein System heißt vollständig beobachtbar, wenn aus dem Verlauf der Ausgangsgröße $v(t)$ und der Eingangsgröße $y(t)$ sämtliche Zustandsgrößen gewonnen werden können. Für die praktische Auswertung ist zu untersuchen, ob die Beobachtbarkeitsmatrix \\ + $\textbf{Q}_B = + \left[ + \begin{array}{c} + \textbf{C} \\ + \textbf{CA} \\ + \textbf{CA}^2 \\ + \vdots\\ + \textbf{CA}^{n-1} + \end{array} + \right] + $ + invertierbar ist.\\ + \subsubheading{Regelbarkeit} + Ein system wird als vollständig regelbar bezeichnet, wenn es sowohl vollständig steuerbar als auch vollständig beobachtbar ist.\\ + Wenn das System nicht vollständig regelbar ist können nicht alle Pole frei gewählt werden, der nicht regelbare Teil wird dann oft als Störung betrachtet. + + \subheading{Pole des ZRkreis} + %\textcolor{magenta}{\textit{Also irgendwie weiß ich noch nicht so recht, wie ich das hier am besten schreibe.}}\\ + + \ccontent{\includegraphics[height=8.1cm]{PolVorg.png}} + Spezielle Polanordnung: ein reeller Pol($\rightarrow$ schwingfähig) und ein konj. komplexes Polpaar mit gleichem Realanteil wie reeller Pol. Der Reelle Pol überdekt die konj. komplexen Pole weswegen es keine Überschwinger gibt. \\ + Wenn Polvorgabe zu langsam $\rightarrow$ auf das Vorzeichen der Reglerkoeffizienten $k_1, k_2,...$ achten. Negative Koeffizienten \textit{können} ein Indiz für ungünstige Polvorgabe sein. Wenn aber eine instabile Strecke stabilisiert werden soll sind negative Koeffizienten normal.\\ + Beobachterpole um Faktor 2...10 schneller als Reglerpole. + + % \subheading{Separationstheorem} + % \textcolor{magenta}{\textit{weiß nicht ob das Thema für die Klausur relevant ist}} + + \subheading{ZR plus I-Anteil} + Um die bleibende Regelabweichung bei konsatnem Sollwert und bei Störgrößen zu vermeiden wird I-Anteil eingefügt. Es soll über $w(t) - v(t)$ also Sollwert - Istwert ingetriert werden. Für schönere Vorzeichen wird dies umgedreht: $v(t) - w(t)$. \\ + $\dot{x}_I(t) = v(t) - w(t)$\\ + $\rightarrow$ Reglerentwurf:\\ + Ordnung um 1 erhöht (Erweiterung des Reglers nicht des Beobachters)\\ + $\rightarrow$ Wahl von $\ell$?\\ + \begin{itemize} + \item nicht für sationäre Genauigkeit benötigt (wegen I-Anteil) + \item $\ell$ produziert eine reelle Nullstelle in $G_w(s)$: $s_0 = -\frac{k_I}{\ell} \rightarrow$ ein Freiheitsgrad + \item häufig: Wahl von $\ell$ so, dass ein Pol von $\tilde{N}(s)$ im Führungsverhalten $G_W(s)$ kompensiert wird. + \end{itemize} + Integrator-Zustand:\\ + $\dot{x}_I(t) = v(t) - w(t) = \textbf{Cx}(t) - w(t)$\\ + Zustandsdarstellung der Regelstrecke: + \begin{flalign*} + \left[ + \begin{array}{c} + \dot{\textbf{x}}(t) \\ + \dot{x}_I(t) \\ + \end{array} + \right] + &= + \underbrace{ + \left[ + \begin{array}{cc} + \textbf{A} & 0 \\ + \textbf{C} & 0 \\ + \end{array} + \right]}_{\textbf{A}_{I}} + \left[ + \begin{array}{c} + \textbf{x}(t) \\ + x_I(t) \\ + \end{array} + \right] + + \underbrace{ + \left[ + \begin{array}{c} + \textbf{B} \\ + 0 \\ + \end{array} + \right]}_{\textbf{B}_{I}} + y(t) + + \left[ + \begin{array}{c} + \textbf{0} \\ + -1 \\ + \end{array} + \right] w(t) & + \end{flalign*} + $det(s\textbf{I} - \textbf{A}_I + \textbf{B}_I\textbf{K}_I) = \tilde{N}(s)$\\ + \textbf{Festlegung der Sollwertgewichtung $\ell$}: \\ + Bisher wurde $\ell$ so gewählt, dass die Regelung stationär genau ist. Dies wird jetzt durch den I-Anteil gewährleistet. Somit ist der Faktor $\ell$ ein freier Parameter. Man kann diesen entweder zu Null setzten oder damit eine Nullstelle im Führungsverhalten platzieren.\\ + \ccontent{\includegraphics[height=3cm]{NstFuehrungsverhalten.png}} + Stellsignal: $y(t) = -(k_1x_1(t) + ... + k_nx_n(t) $ \textcolor{red}{$k_I x_I(t)$} $ + \ell w(t))$\\ + $G(s) = \frac{Vorw}{1+R"uckw} = \frac{\ell + \frac{1}{s}k_I}{...} = \frac{s\ell + k_I}{s(...)} \rightarrow$ Nst. bei $s_0 = -\frac{k_I}{\ell}$ \\%in $G_w(s)$ + $det(s\textbf{I} - \textbf{A}_I + \textbf{B}_I\textbf{K}_I) \overset{!}{=} \tilde{N}(s) = \underset{n}{\overset{i = 1}{\prod}} (s-s_{\infty,i})$\\ + \textcolor{red}{\textbf{Nicht möglich:}} $l = \frac{1}{\textbf{C}(\underbrace{s\textbf{I} - \textbf{A}_I + \textbf{B}_I\textbf{K}_I}_{nichtinvertrbar, det() = 0} )^{-1} \textbf{B}_I}$\\ + Das Führungsverhalten ist mit I-Anteil gleich zu dem ohne, wenn der hinzugekommene Pol wegkompensiert wird. $\rightarrow \ell_{ohne~I-Anteil} = \ell_{mit~I-Anteil}$, wenn $s_0 = \frac{-k_I}{\ell} \overset{!}{=}$ zusätzlicher Pol (den dann auch in Gleichung weglassen). \\ + \textbf{Zusammenfassung ZR mit I-Anteil:} + \begin{itemize} + \item Idee: Durch Hinzunehmen eines Zustands, der aus der Differenz von Soll- und Istwersignal gespeist wird, lässt sich erreichen, dass im Führungs- und Störverhalten keine bleibende Regelabweichung auftritt. + \item Vorteilhaft: einfach Erweiterung, zielt auf die Wirkung in der Regelgröße ab, Standard-Beobachter kann verwendet werden. + \item Nachteilig: Beobachter schätzt die Zustände nicht korrekt. + \item Anzahl der vorzugebenden Pole für Zustandsregler: $n + 1$ ($n$ = Systemordnung) + \item Standard-Beobachter (Ordnung = Streckenordnung $n$) + \end{itemize} + \subheading{Störmodell} + Verbesserung der Beobachtung von Zustandsgrößen durch ein Modell einer konstanten Störung. Für konstante oder sich langsam ändernde Störungen wird ein Integrator als Störmodell eingesetzt. \\ + \ccontent{\includegraphics[height=4.7cm]{Stoermodell.png}} + \textcolor{orange}{Störung ist nunmehr ein modelleirter Zustand für den Beobachterentwurf (kein Eingang)} + \begin{flalign*} + \dot{\textbf{x}}_S(t) + &= + \underbrace{ + \left[ + \begin{array}{cc} + \textbf{A} & \textcolor{orange}{\textbf{G}} \\ + 0 & 0 \\ + \end{array} + \right]}_{\textbf{A}_S} + \textbf{x}_S(t) + + \left[ + \begin{array}{c} + \textbf{B} \\ + 0 \\ + \end{array} + \right] + y(t) &\\ + \textbf{C}_S &= [~\textbf{C}~\textcolor{orange}{\textbf{G}}~] \textcolor{orange}{\textbf{G}\textbf{ je nach dem wo die Störung eingeht}}&\\ + \textbf{x}_S(t) &= \left[ + \begin{array}{c} + \textbf{x}(t) \\ + x_S(t) \\ + \end{array} + \right] + \end{flalign*} + Wegen des Integrators hat der Beobachter jetzt einen Pol mehr. \\ + In der Literatur gibt es zwei Möglichkeiten zur Unterdrückung konstanter Störungen: + \begin{itemize} + \item ZR mit I-Anteil-Erweiterung + \item ZR ohne I-Anteil-Erweiterung mit Beobachter mit Störmodell und mit Störgrößenaufschaltung + \end{itemize} + Berechnung einer geeigneten Störgrößenaufschaltung:\\ + Annahme: Beobachter mit Störmodell liefert korrekte Zustände, bzw. schätzt asymptotisch korrekt. $\rightarrow~ k_s$-Bestimmung ohne Beobachter (d.h. wie diskrete Messung des Zustandes \textbf{x}(t) und der Störung z(t)!) \\ + $\hat{\textbf{x}}(t) = \textbf{x}(t)$; $x_s(t) = z(t)$ geschätzte = tatsächliche Störung. \\ + Wahl von $k_s$ für stationär genaues Störverhalten $G_z(0) = 0$?\\ + $G_z(s) = \frac{V(s)}{Z(s)}$ mit $v = \textbf{Cx}$\\ + Strecke: $\dot{\textbf{x}} = \textbf{Ax} + \textbf{B}y + \textbf{G}z$\\ + Stellsignal: $y = -\textbf{Kx} + \ell w + k_sz$\\ + $\rightarrow \dot{\textbf{x}} = (\textbf{A} - \textbf{BK})\textbf{x} + \textbf{B}\ell w + (\textbf{G} - \textbf{B}\textcolor{red}{k_s})z$\\ + \ccontent{\includegraphics[width = \columnwidth]{Stoergroessenaufschaltung.png}} + $\rightarrow$ Störübertragungsfunktion:\\ + $G_z(s) = \textbf{C}(s\textbf{I} - \textbf{A} + \textbf{BK})^{-1}(\textbf{G} - \textbf{B}\textcolor{red}{k_s})$\\ + $G_z(0) = \textbf{C}(- \textbf{A} + \textbf{BK})^{-1}(\textbf{G} - \textbf{B}\textcolor{red}{k_s}) \overset{!}{=} 0$\\ + $G_z(0) = \textbf{C}(- \textbf{A} + \textbf{BK})^{-1} \textbf{G} - \textbf{C}(- \textbf{A} + \textbf{BK})^{-1} \textbf{B}\textcolor{red}{k_s}) \overset{!}{=} 0$\\ + $\textcolor{red}{k_s} = \frac{\textbf{C}(- \textbf{A} + \textbf{BK})^{-1} \textbf{G}}{\textbf{C}(- \textbf{A} + \textbf{BK})^{-1} \textbf{B}}$ + \textbf{Zusammenfassung ZB mit Störmodell: }\\ + \begin{itemize} + \item Idee: Das Streckenmodell im Beobachter wird um einen Störungs-Zustand erweitert. Dessen Wirkung ist an der Stelle zu modellieren, an der die Störung tatsächlich eingreift. Bei mechanischen Systemen ist dies in der Regel die Summationsstelle der angreifenden Kräfte. + \item Vorteilhaft: Die Zustände werden richtig geschätzt, auch wenn Störungen angreifen. Durch Kombination mit einer Störgrößenaufschaltung lässt sich erreichen, dass nicht nur die Zustände korrekt geschätzt werden, sondern dass auch keine bleibende Regelabweichung im Führungs- und Störverhalten auftritt. + \item Nachteilig: In vielen Fällen etwas aufwenigere Modellierung und Rechnung + \item Anzahl der vorzugebenden Pole für Beobachter: $n + 1$ ($n$ = Streckenordnung) + \end{itemize} + + \subheading{allgemeinere Regleranteile} + \subsubheading{Doppelter I-Anteil im ZR} + Zur Vermeidung von Schleppfehlern wird der Zustandsregler mit einem doppelten I-Anteil erweitert. \\ + \ccontent{\includegraphics[height=3cm]{Doppelintegrator.pdf}} + \begin{flalign*} + \left[ + \begin{array}{c} + \dot{\textbf{x}}(t) \\ + \dot{x}_{I,1}(t) \\ + \dot{x}_{I,2}(t) \\ + \end{array} + \right] + &= + \underbrace{ + \left[ + \begin{array}{ccc} + \textbf{A} & 0 &0\\ + \textbf{0}&0&1\\ + \textbf{C} & 0 &0\\ + \end{array} + \right]}_{\textbf{A}_{I^2}} + \left[ + \begin{array}{c} + \textbf{x}(t) \\ + x_{I,1}(t) \\ + x_{I,2}(t) \\ + \end{array} + \right] + + \underbrace{ + \left[ + \begin{array}{c} + \textbf{B} \\ + 0 \\ + 0\\ + \end{array} + \right]}_{\textbf{B}_{I^2}} + y(t) &\\ + &+ + \left[ + \begin{array}{c} + \textbf{0} \\ + 0\\ + -1 \\ + \end{array} + \right] w(t) & + \end{flalign*} + Hat aber meist Überschwinger. + + \subsubheading{Sinusförmige Führungs- oder Störsignale} + Bei Sinusförmigen Sollwerten- und/oder Störverläuften (50 Hz Netzfrequenz als Störung) muss besonders geregelt werden um Amplituden und Phasenfehler zu vermeiden. Ist die Frequenz bekannt, kann ein Regleranteil/Beobachter verwendet werden, der genau diese Frequenz enthält. Hier die Beobachterergänzung: \\ + \begin{flalign*} + \left[ + \begin{array}{c} + \dot{\textbf{x}}(t) \\ + \dot{x}_{sin,1}(t) \\ + \dot{x}_{sin,2}(t) \\ + \end{array} + \right] + &= + \underbrace{ + \left[ + \begin{array}{ccc} + \textbf{A} & \textbf{G} &0\\ + \textbf{0}&0&1\\ + \textbf{0} & -\omega_0^2 &0\\ + \end{array} + \right]}_{\textbf{A}_{sin}} + \left[ + \begin{array}{c} + \textbf{x}(t) \\ + x_{sin,1}(t) \\ + x_{sin,2}(t) \\ + \end{array} + \right] + + \underbrace{ + \left[ + \begin{array}{c} + \textbf{B} \\ + 0 \\ + 0\\ + \end{array} + \right]}_{\textbf{B}_{sin}} + y(t) &\\ + &+ + \left[ + \begin{array}{c} + \textbf{0} \\ + 0\\ + -1 \\ + \end{array} + \right] w(t) & + \end{flalign*} + \textbf{C}$_{sin} = [\textbf{C}~\textbf{G}~0]; \textbf{H}_{sin} = [h_1;~h_2;~h_3;~h_{sin,1};~h_{sin,2}]$ \textbf{G}, je nach dem wo die Störung wirkt. \textbf{H} wird wie immer bestimmt: $det(s\textbf{I} - \textbf{A} + \textbf{HC}) = \Delta(s)$\\ + \ccontent{\includegraphics[height=3.5cm]{SinusStoermodellStoer.pdf}} \textit{Als Regleranteil auf Seite 10} + + \subheading{Frequenzbereichsentwurf} + \begin{itemize} + \item alternativer Entwurfsweg: ZR + Beobachter gemeinsam (meist einfacherer Entwurfsgang) + \item basierend auf Übertragungsfunktionen + \item Zustandsregler mit reduziertem Beobachter, falls I-Anteil: enspricht Erweiterung des Beobachters plus Störgrößenaufschaltung + \item ggf. einfachere Realisierung + \item Interne Größen des Beobachters können aber nicht einfach herausgeführt werden + \end{itemize} + \ccontent{\includegraphics[height=4cm]{ZRKreduzBeobFarbe2.png}} + \ccontent{\includegraphics[height=4cm]{FrequenzbereichsdarstellungZRKreduzBeobFarbe1_2.png}} + Übertragungsfunktionen $G_y(s)$ und $G_v(s)$ entstammen dem Beobachter incl. Zustandsregler. Dynamik des reduzierten Beobachters durch charakteristische Matrix \textbf{F}, Eigenwerte durch die Polvorgabe vorgegeben: $det(s\textbf{I} - \textbf{F}) = \Delta (s) = 0$\\ + Deswegen haben $G_y(s)$ und $G_v(s)$ beide den Nenner $\Delta (s)$ und können geschrieben werden als: $G_y(s) = \frac{Z_Y(s)}{\Delta(s)}~~G_v(s) = \frac{Z_V(s)}{\Delta(s)}$ + \subsubheading{Entwurfsgleichung im Frequenzbereich} + \colorbox{yellow}{$N_S(s)N_R(s) + Z_S(s)Z_V(s) = \Delta(s) \tilde{N}(s)$}, bekannt sind die Polynome $N_S(s)$ und $Z_S(s)$ der Streckenübertragungsfunktion und die vorgegebenen Polynome $\Delta(s)$(Beobachterpole Grad = Grad d. StreckenOrd -1 (-1 weil reduziert)) sowie $\tilde{N}(s)$(Pole der Zustandsregelung Grad = Grad d. StreckenOrd). Den Rest bekommt man durch Koeffizientenvergleich. \colorbox{yellow}{$Z_Y(s) = N_R(s) - \Delta(s)$} hat maximalen Polynomgrad $n_{Beob} - 1$. Wenn $N_R(0) = 0$, hat der ZR einen I-Anteil. Sollwertgewichtung: \colorbox{yellow}{$\ell = \frac{\tilde{N}(0)}{Z_S(0)}$} (ist beim Frequenzbereichsentwurff mit I-Anteil NICHT frei wählbar). Bei $N_S(s)$ ist wichtig, dass in folgender Form: \colorbox{yellow}{$N_S(s) = \textcolor{red}{1}s^2 + c_1 s + c_0$. $Z_V = as +b$ $N_R = s+c$} haben die Beobachterordnung (für Koeffizientenvergleich).\\ + + \subsubheading{Strukturen} + \begin{itemize} + \item 2-kanalig: wegen Vermeidung von Beobachtungsfehlern bei Stellsignalbegrenzung, keine Probleme bei getrennt volllaufenden Integratoren $G_Y$, $G_V$. $G_0 = G_y + G_s\cdot G_v$ + \item 1-kanalig: Übertragungsfunktion einfach zu rechnen, \textbf{nicht} für Realisierung (weben Stellsignalbegrenzung). $G_0 = \frac{Z_S \cdot Z_V}{N_S \cdot N_R}$ + \end{itemize} + \ccontent{\includegraphics[height=3.5cm]{HerleitungEinschlStruktur1.png}} + \ccontent{\includegraphics[height=6cm]{HerleitungEinschlStruktur2.png}} + $G_w(s) = \ell \frac{Z_S(s)}{\tilde{N}(s)}~\rightarrow$ Im Führungsverhalten tritt die Beobachterdynamik nicht auf. + $G_w(s)$ mit Beobachter = $G_w(s)$ ohne Beobachter.\\ + $G_z(s) = \frac{V(s)}{Z(s)}$ über Einkanalige Struktur ermitteln. $G_{yw} = \frac{Y(s)}{W(s)} = \frac{G_w}{G_s}$\\ + $G_o = G_y + G_s\cdot G_v$ offener Regelkreis zB für nichtlin. Dauerschw. + \subsubheading{I-Erweiterung im Frequenzbereich} + Entwurfsgleichung: $N_S(s)\underbrace{N_R(s)}_{\textcolor{Green}{+1}} + Z_S(s)\underbrace{Z_V(s)}_{\textcolor{Green}{+1}} \overset{!}{=} \underbrace{\Delta(s)}_{\textcolor{orange}{+1}} \tilde{N}(s)$ + \begin{itemize} + \item Erhöhung der Beobachterordnung um \textcolor{orange}{1} (entspricht Störmodell = Integrator) + \item \textcolor{Green}{+2} zusätzliche Koeffizienten in $Z_V(s)$ und $N_R(s)$ + \item eine Bedingung frei $\rightarrow$ Integratoranteil in $N_R(s)$! + \end{itemize} + Pole: + \begin{itemize} + \item für Regelung: $n$ Pole vorgegeben + \item für Beobachter $n - n_{Anzahl Messgr"ossen} + 1_{I-Anteil}$ \textcolor{Purple}{Pole} zu vergeben + \end{itemize} + $N_S(s)(s^{\textcolor{Purple}{n_B}} + ... + a_1s+a_0) + Z_S(s)(b_{\textcolor{Purple}{n_B}}s^{\textcolor{Purple}{n_B}} + ... + b_1s+b_0) = \Delta(s) \tilde{N}(s)$ wobei $a_0 = 0$ wegen I-Anteil. \\ + \textbf{Sinus-Erweiterung im Frequenzbereisentwurf:}\\ + $n_s$ Regelungspole vorzugeben $\tilde{N}(s)$\\ + $n_B = n_s \underbrace{-1}_{\underline{eine}~Messgr"osse} \underbrace{+2}_{Sinusmodell~hat~2~Zust"ande}$ \\ + Beobachtungspole vorzugeben $\Delta(s)$\\ + Ansatz für\\ + \begin{flalign*} + N_R(s) &= s^{n_B} + a_{n_B-1}s^{n_B-1}+...+a_0 &\\ + &= \underbrace{(s^2+w_0^2)}_{2~Pole~SinSt"oermodell} (s^{n_B-2} + \alpha_{n_B-3}s^{n_B-3}+...) &\\ + \end{flalign*} + + \heading{Regelkreise mit Stellsignalbegrenzungen} + \textbf{Typische Probleme:} + \begin{itemize} + \item Langsamere Streckenreaktion als ohne Betrenzung. + \item Überlaufender Integralanteil bei Reglern mit I-Anteil + \end{itemize} + Maximal/Minimal mögliches Ausgangssignal: $v_{min/max} = w_{min/max}\cdot G_s(0) + z_{min/max}$\\ + Gibt keine Führungsübertragungsfunktion, weil nicht linear. + \subheading{Stellsignalbegrenzung bei ZRK mit Beob.} + Wird das Stellsignal begrenzt, kann der I-Anteil sehr hohe Werte annehmen (Integral windup). Dagegen kann + \begin{itemize} + \item der Integrator angehalten (Clamping) werden oder + \item die Differenz aus begrenzter und unbegrenzter Stellgröße auf den Integrator Rückgeführt werden (Begrenzungs-Beobachter). + \end{itemize} + \ccontent{\includegraphics[height=6cm]{StellSigBegr.png}} + Oben: Anhalten, Unten: Rückführung. Anti-Windup in Lila, PI in Grün. Für den Gain-Block Unten 0.15*u: Daumenregel bei PID-Reglern: $Kb \approx \sqrt{\frac{1}{TN\cdot TV}}$. Beide Lösungen sind passiv und nur aktiv wenn das Stellsignal in die Begrenzung geht. Für ZR mit Beobachter muss die Begrenzung auch beachtet werden, sonst enstehen Beobachtungsfehler (Beobachtwer begrenztes Stellsignal zuführen). + \ccontent{\includegraphics[height=6cm]{ZRStellBegrBeob.png}} + Zustandsregler mit I-Anteil und Einheitbeobachter + + \subheading{nichtlineare Dauerschwingung} + Regelkreise mit Stellbegrenzung $\rightarrow$ in vielen Fällen Dauerschwingungen durch Zusammenspiel nichtlinearer Kennlinien und linearen, dynamischen Systemanteilen. Schwingungsbedingung: \\ + $G_o(j \underbrace{\omega_x}_{ges. Kreisfrequenz}) \cdot K_N(\underbrace{v_o}_{ges. Amplitude}) = -1 \rightarrow $\\ + \colorbox{yellow}{$\underbrace{G_o(j\omega_x)}_{Ortskurve} = \underbrace{-1/K_N(v_o)}_{neg.~inv.~Beschreibungsfunktion}$}\\ + $G_o(s) = \textbf{K}\underbrace{{(s\textbf{I} - \textbf{A})}^{-1}} \textbf{B}$\\ + Pole da, wo ungeregelte Strecke die Pole hat. \\ + $G_o(s) = \frac{\tilde{N}(s)}{N_s(s)}-1 = F_o(s) -1$ $~~F_o(s) = G_0 +1$ Frequenzgangsfunktion die untersucht werden muss (linearer Systemanteil für die harmonische Balance) + + \subheading{Phasenkriterium bei begrenzen Stellsignalen} + Im Bodediagramm von $F_o(j\omega)$ schauen, ob die Phase oberhalb von -140$\degree$ verläuft, dann kein Überschwingen. Bzw. in Ortskuve: \\ + \ccontent{\includegraphics[height=2.7cm]{OrtskurveDauerschw.png}} + Wenn im Bodediagramm die -140$\degree$ geschnitten werden: \\ + 1. die Pole langsamer wählen bis der Phasenverlauf nicht mehr die die 140$\degree$ schneidet, oder:\\ + 2. ein Zusatznetzwerk dimensionieren (nur wenn 1. nicht geht).\\ + \ccontent{\includegraphics[height=4cm]{DauerSchwZusatzNetzwerk.png}} + Zusatznetzwerk $G_1(s) = \frac{\tilde{N}_1(s)}{N_1(s)}-1$ hebt Phase von $G_{ges}~=~G_{o1}$. $G_{o1} = \frac{\tilde{N}(s) N_1(s)}{\tilde{N}_1(s) N(s)} -1 = F_o(s)F_1^{-1}(s) -1 \rightarrow$ $\tilde{N}_1(s)$ und $N_1(s)$ gleicher Grad und Koeffizient bei größter Potenz von s gleich 1 um eine Anordnung ohne Durchgriff zu erhalten. + + \heading{Ideales Folgen} + Ziel: $w(t) = v(t) ~\forall ~t$\\ + \ccontent{\includegraphics[height=1.5cm]{IFSystem.png}} + $V(s) = Y(s)G_s(s) = W(s) \underbrace{G_x(s) G_s(s)}_{\overset{!}{=}1} \overset{!}{=} W(s)$ + \colorbox{yellow}{$G_x(s) = \frac{1}{G_s(s)}$}\\ + Theoretisches Ergebnis: Vorsteuerung mit der \textit{inversen Regelstrecke} löst die Aufgabe des idealen Folgens. (ohne Linearisierung) \\ + Aber: + \begin{itemize} + \item \textbf{Realisierbarkeit?} Regelstrecken haben keinen Durchgriff: Zählergrad $m_s < $ Nennergrad $n_s$. $\rightarrow$ Inverses System hat damit $n_s$ Nullstellen und $m_s$ Pole ist also \textbf{nicht realisierbar}. \\ \textcolor{orange}{Lösung: Realisierbares Stellglied reslutiert, wenn w(t) hinreichend oft differenzierbar ist.} + \item \textbf{Was ist bei nicht minimalphasigen Regelstrecken?} Regelstrecke hat mind. eine Nst. außerhalb des Stabilitätsgebiets (kontinuierliches System: in der rechten s-Halbebene; zeitdiskret: Außerhalb des Einheitskreises). $\rightarrow$ Inverses System $G_x(s)$ hat mind. einen Pol außerhalb des Stabilitätsgebiets.\\ \textcolor{orange}{Lösung: wähle eine andere Ausgangsgröße (\textit{flachen} Ausgang suchen)} + \end{itemize} + Wenn $w(t)$ eine Funktion mit sin/cos ist, kann asymptotisch ohne Sin-Modell eingeregelt werden. + \subheading{Für lineare zeitinvariante Systeme erster Ordnung} + Beispiel: $PT_1$\\ + $\dot{x}(t) = Ax(t) + By(t)$; $v(t) = C x(t)$ mit $A, B, C$ als skalare Konstanten ungleich Null. $G(s) = \frac{V(s)}{Y(s)} = \frac{CB}{s-A}$\\ + \textbf{1. Schritt:} setze $v(t) = C x(t) \overset{!}{=} w(t) \rightarrow$ kann nicht nach $y(t)$ augelöst werden. \\ + \textbf{2. Schritt:} Ausgangsgleichung nach $t$ differenzieren:\\ $\dot{v}(t) = C \dot{x}(t) = C(Ax(t) + By(t)) = CAx(t) + CBy(t) \overset{!}{=} \dot{w}(t)$; Auflösen nach \colorbox{yellow}{$y(t) = \frac{\dot{w}(t) - CAx(t)}{CB}$ Stellgesetz} 'kompensierende Rückführung'. + Schritt 1: Kompensation der Streckendynamik:\\ + \ccontent{\includegraphics[width = 0.7\columnwidth]{IdFoSchr1.png}} + Schritt 2: Ergänzung durch einen Regler:\\ + \ccontent{\includegraphics[width = 0.3\columnwidth]{IdFoSchr2.png}} + %Alternativ in einem Bild handschriftlich: + %\ccontent{\includegraphics[width = \columnwidth]{IdFoSchr12.png}} + Bei einer Störung, gäbe es aber eine bleibende Regelabweichug. \\ + \textbf{Achtung!} Wenn man einen Beobachter braucht, wir die kompensierende Rückführung natürlich an den Beobachter angeschlossen! + + \subheading{Für Systeme höherer Ordnung} + Gleiche Vorgehensweise wie oben. Wenn aber $\textbf{CB} = 0$ wird wieder differenziert, solang bis das Produkt vor $y(t) \neq 0$ ist. Man erhält dann \colorbox{yellow}{$y(t) = \frac{w^{(r)}(t) - \textbf{CA}^rx(t)}{\textbf{CA}^{r-1}\textbf{B}}$} mit $r$-facher Integratorenkette zwischen $w^{(r)}(t)$ und $v(t)$.\\ + Schritt 1: Kompensation der Streckendynamik:\\ + \ccontent{\includegraphics[width = 0.75\columnwidth]{IdFo2Schr1.png}} + so ensteht eine $I_r$-Kette: $\frac{1}{s^r}$ dies entspricht einer Zustandsrückführung mit:\\ + 'Polvorgabe': $n_B = n$ Streckenordnung, $n_R = r +1$ (Für ein $I_r$-System, r Pole nach Null, n-r Pole = Nst. der Regelstrecke $G_s(s)$)(+1 für I-Anteil). Beob.Pole und Regelungspole tragen zur Ausregelzeit bei also alle berücksichtigen. \\ + \textbf{Beispiel (A4.2):} Ideales Folgen bei $PD-T_3$-Strecke\\ + geg: $G(s) = \frac{\textcolor{Green}{0.5}s + \textcolor{Green}{1}}{s^3 + \textcolor{Purple}{2}s^2 + \textcolor{Purple}{2}s + \textcolor{Purple}{1}}$ ZR mit I-Anteil gewünscht um sprungförmige Störungen asymptotisch auszuregeln. Alle vorzugebenden Pole auf -4.\\ + Zustandsraumdarstellung z.B. in RNF (2. kanonische Struktur): + \begin{flalign*} + \dot{\textbf{x}} (t) + &= + \underbrace{ + \left[ + \begin{array}{ccc} + \textcolor{orange}{0} & \textcolor{orange}{1} &\textcolor{orange}{0}\\ + \textcolor{orange}{0}&\textcolor{orange}{0}&\textcolor{orange}{1}\\ + \textcolor{Purple}{-1} & \textcolor{Purple}{-2} &\textcolor{Purple}{-2}\\ + \end{array} + \right]}_{\textbf{A}} + \textbf{x}(t) + + \underbrace{ + \left[ + \begin{array}{c} + \textcolor{RoyalBlue}{0} \\ + \textcolor{RoyalBlue}{0} \\ + \textcolor{RoyalBlue}{1}\\ + \end{array} + \right]}_{\textbf{B}} + y(t);~ + v(t) = \underbrace{\left[ + \begin{array}{c c c} + \textcolor{Green}{1} & + \textcolor{Green}{0.5} & + 0\\ + \end{array} + \right]}_{\textbf{C}} \textbf{x}(t) &\\ + \end{flalign*} + \textcolor{orange}{Integratorenkette}, \textcolor{RoyalBlue}{immer gleich}. \\ + Kompensationsrückführung nach zweimal differenzieren: $y(t) = \frac{\ddot{w}(t) - \textbf{CA}^2x(t)}{\textbf{CAB}}$, nächster Schritt: Reglerentwurf für $I_r = I_2$-System\\ + \ccontent{\includegraphics[width = 0.4\columnwidth]{I2System.png}} + \begin{flalign*} + \dot{\textbf{x}}_{I_2} (t) + &= + \left[ + \begin{array}{cc} + 0 & 1\\ + 0&0\\ + \end{array} + \right] + \textbf{x}_{I_2}(t) + + \left[ + \begin{array}{c} + 0 \\ + 1 \\ + \end{array} + \right] + y_{I_2}(t) &\\ + \end{flalign*} + Zustandsreglerentwurf besonders einfach:\\ + ZR mit I-Anteil: + \begin{flalign*} + \textbf{A} + &= + \left[ + \begin{array}{cc} + \textbf{A}_{I_2} & \textbf{0}\\ + \textbf{C}_{I_2}&0\\ + \end{array} + \right] = + \left[ + \begin{array}{ccc} + 0 & 1 & 0\\ + 0&0&0\\ + 1&0&0\\ + \end{array} + \right]; ~ + \textbf{B} = + \left[ + \begin{array}{c} + \textbf{B}_{I_2} \\ + 0 \\ + \end{array} + \right] = + \left[ + \begin{array}{c} + 0 \\ + 1 \\ + 0 \\ + \end{array} + \right] &\\ + \end{flalign*} + Polvorgabe: 3 Pole $\rightarrow (s+4)^3 = s^3 + 12s^2 + 48s +64$\\ + Koeffizientenvergleich mit: \\ + $det(s\textbf{I}- \textbf{A} + \textbf{Bk}) = det\left( + \begin{array}{ccc} + s & -1 & 0\\ + k_1&s+k_2&k_I\\ + -1&0&s\\ + \end{array} + \right) = s^3 + k_2s^2 + k_1s + k_I$\\ + + mit $\textbf{k} = [k_1~k_2~k_I]$ ergibt $k_2 = 12,~ k_1 = 48,~k_I = 64$\\ + \textcolor{red}{Achtung beim Anschließen des ZR: Regler darf nur die Abweichung der Trajektorien sehen}\\ + ZR: $y_R(t) = k_1(w(t) - v(t)) + k_2(\dot{w}(t)- \dot{v}(t)) + k_I\int^t w(t)-v(t)dt$ mit $\dot{v}(t) = \textbf{CAx}(t)$ \\ + Hier muss $v(t)$ ggf abgeleitet werden. Bsp:\\ + $\dot{x}_1 = x_2; \dot{x}_2 = x_3; \dot{x}_3 = 5x_2 +y; v = [2~5~0]x; \dot{v} = [2~5~0]\dot{x}= [0~2~5]x$ + + \subheading{Altern. Realisierung ohne Kompensationsrückführung} + \textbf{Beispiel:} Gegeben ist eine PDT$_3$-Regelstreckemit der Übertragungsfunktion $G_s(s) = \frac{a + bs}{c +ds+es^2+fs^3}$. Als ideales Vorsteuersignal erhält man somit + \begin{flalign*} + Y_v(s) &= W(s) \frac{1}{G_s(s)} = W(s) \cdot \underbrace{\frac{c+ds+es^2+fs^3}{a + bs}}_{geht~nicht~weil~mehr~Nst.~als~Pole} \\ + &= W(s)(c++ds+es^2+fs^3)\frac{1}{a + bs} \\ + &= (W(s)c + sW(s) d + s^2W(s) e + s^3W(s)f)\frac{1}{a + bs} &\\ + \end{flalign*} + \begin{flalign*} + y_v(t) &= (w(t) c + \dot{w}(t)d + \ddot{w}(t) e + w^{(3)} (t)f)*g_{PT_1}(t) &\\ + \end{flalign*} + mit der Impulsantwort des $PT_1$-Systems $g_PT_1(t) = \mathscr{L}^{-1} \{ \frac{1}{a+bs}\}$ + \ccontent{\includegraphics[width = \columnwidth]{IdFoOhneKompRueck.png}} + Mit dieser Struktur kann der Regler doch realisiert werden. $G_R(s)$ dient hier zum ausregeln von Störungen. + \ccontent{\includegraphics[width = \columnwidth]{IdFoOhneKomp12.png}} + $\rightarrow G_I(s) = \frac{1}{G_s(s)}$ für Regelstrecke; $\rightarrow G_I(s) = \frac{1}{G_w(s)}$ für die Führungsübertragungsfunktion zu bestimmen + + \subheading{Ein-/Ausgangslinearisierung nichtlinearer Systeme} + Systemtyp: + \begin{flalign*} + \dot{\textbf{x}}(t) &= \textbf{a}(\textbf{x}(t)) + \textbf{b}(\textbf{x}(t)) \cdot \textcolor{Green}{y(t)}~ \textcolor{Green}{\text{ eingangsaffin}} \\ + v(t) &= c(\textbf{x}(t)) &\\ + \end{flalign*} + mit hinreichend häufig differenzierbaren Funktionen \textbf{a}, \textbf{b} und c. + \begin{flalign*} + \dot{v}(t) &= \frac{\partial c(\textbf{x}(t))}{\partial x_1(t)} \cdot \frac{dx_1(t)}{dt} + \frac{\partial c(\textbf{x}(t))}{\partial x_2(t)} \cdot \frac{dx_2(t)}{dt} + ... + \frac{\partial c(\textbf{x}(t))}{\partial x_n(t)} \cdot \frac{dx_n(t)}{dt} \\ + &= \frac{\partial c(\textbf{x}(t))}{\partial \textbf{x}(t)^{T}}\dot{\textbf{x}}(t) \\ + &= \frac{\partial c(\textbf{x}(t))}{\partial \textbf{x}(t)^{T}} (\textbf{a}(\textbf{x}(t)) + \textbf{b}(\textbf{x}(t)) \textcolor{Green}{y(t)}) &\\ + \end{flalign*} + Bsp: $v(t) = x_1(t) + x_2^3(t) \rightarrow \dot{v}(t) = 1\cdot \dot{x}_1(t) + 3x_2^2(t)\cdot \dot{x}_2(t)$\\ + Wenn $\frac{\partial c(\textbf{x}(t))}{\partial \textbf{x}(t)^{T}}\textbf{b}(\textbf{x}(t)) \neq 0$ kann man $\dot{v}(t)$ der zeitlichen Ableigung des Sollwertsignals $\dot{w}(t)$ gleichsetzten und nach dem Stellsignal auflösen. + \begin{flalign*} + y(t) &= \frac{\dot{w}(t) - \frac{\partial c(\textbf{x}(t))}{\partial \textbf{x}(t)^{T}}\textbf{a}(\textbf{x}(t))}{\frac{\partial c(\textbf{x}(t))}{\partial \textbf{x}(t)^{T}}\textbf{b}(\textbf{x}(t))} + \end{flalign*} + Ansonsten muss so lange abgeleitet werden, bis dies der Fall ist. \\ + \textbf{Aufgabe 4.3:}\\ + \begin{flalign*} + \dot{x}_1(t) &= x_2(t) &\\ + \dot{x}_2(t) &= cos(x_1(t)) + y(t) &\\ + v(t) &= sin x_1(t) &\\ + \end{flalign*} + $v(t)$ nicht direkt von $y(t)$ beeinflussbar $\rightarrow \frac{d}{dt}$ + \begin{flalign*} + \dot{v}(t) &= cos(x_1(t)) \cdot \dot{x}_1(t) = cos(x_1(t)) \cdot x_2(t) &\\ + \ddot{v}(t) &= \frac{\partial \dot{v}}{\partial x_1} \dot{x}_1(t) + \frac{\partial \dot{v}}{\partial x_2} \dot{x}_2(t) &\\ + &= -sin(x_1(t)) x_2(t) \dot{x}_1(t) + cos(x_1(t)) \dot{x}_2(t) &\\ + &= -sin(x_1(t)) x_2(t) x_2(t) + cos(x_1(t))(cos(x_1(t))+y(t)) \overset{!}{=}\ddot{w}(t) &\\ + y(t) &= \frac{\ddot{w}(t) + sin(x_1(t))x_2^2(t) - cos(x_1(t))^2}{cos(x_1(t))} + \end{flalign*} + 'kompensierende Rückführung' $\rightarrow$ 2-fache Integratorenkette\\ + Singularität: $cos(x_1(t)) = 0$ hierfür ist die exakte Linearisierung nicht möglich! + \subheading{Festwertregelung von nichtlinearen Systemen} + \begin{itemize} + \item Wenn ideales Folgen inklusive Trajektroienplanung nicht nötig ist (in vielen Anwendungen) + \item Nichtlineare Regelung/Zustandsrückführung um feste Arbeitspunkte würde genügen + \item Trajektorienplanung weglassen + \end{itemize} + + \subheading{Nulldynamik} + Nulldynamik beschreibt, was im Inneren eines Systems mit Kompensationsrückführung passiert, wenn die Ausgangsgrößen Null sind, bzw. wenn das Innere instabil wird. Wie minimalphasigkeit bei linearen Systemen wo alle Nst. der Streckenübertragungsfunktion im Stabilitätsgebiet liegen müssen. \\ + Wenn Systemordnung = Differenzenordnung gibt es keine instabile Nulldynamik.\\ + \textbf{Beispiel}: \\ + \begin{flalign*} + \dot{\textbf{x}}(t) + &= + \left[ + \begin{array}{cc} + -1 & 1\\ + -1&0\\ + \end{array} + \right] + \textbf{x}(t) + + \left[ + \begin{array}{c} + 1 \\ + a \\ + \end{array} + \right] + y(t); ~~~~ + G_s(s) = \frac{s+a}{s^2 +s +1} \\ + v(t) &= [1~0] \textbf{x}(t) \\ + \dot{v}(t) &= [1~0] + \left[ + \begin{array}{cc} + -1 & 1\\ + -1&0\\ + \end{array} + \right] + \textbf{x}(t) + + [1~0] + \left[ + \begin{array}{c} + 1 \\ + a \\ + \end{array} \right] y(t) \\ + &= [-1~1]\textbf{x}(t) + 1\cdot y(t) = -x_1(t) + x_2(t) + y(t) \overset{!}{=} \dot{w}(t)\\ + y(t) &= \frac{\dot{w}(t) + x_1(t) - x_2(t)}{1} + &\\ + \end{flalign*} + Welche Dynamik existiert im System 'Strecke' mit Kompensationsrückführung wenn Ausgangssignal $v(t)$ für alle t gleich (identisch) Null ist?\\ + $v(t) = 0;~ \dot{v}(t) = 0; ~ \ddot{v}(t) = 0; ...$\\ + Im Beispiel: $v(t) = x_1(t) = 0$ und $\dot{w}(t) = 0$\\ + Was macht $x_2(t)$? ($y(t)$ in $\dot{x}_2$ einsetzten)\\ + \begin{flalign*} + \dot{x}_2(t) &= -1x_1(t) + a \underbrace{(\dot{w}(t) + x_1(t) - x_2(t))}_{y(t),~Kompensationsr"uckf"uhrung} \\ + \textcolor{Purple}{\dot{x}_2(t)} &\textcolor{Purple}{= a(-x_2(t))} ~~\textcolor{Purple}{Nulldynamik}&\\ + \end{flalign*} + Stabil, wenn $a>0$ + \begin{itemize} + \item Hat ein System Nullstellen, so gibt es Systemtrajektorien, bei denen das Ausganssignal identisch null ist, während das Eingangssignal und ein Teil des Zustandsvektors ungleich null sind. Diese Dynamikanteile bezeichnet man auch als 'interne Dynamik'. + \item Hat ein System nichtminimalphasiges Übertragungsverhalten, so ist die 'interne Dynamik' instabil. + \item Wenn Nullstellen von $G_s(s)$ kleiner Null sind also links liegen ists stabil + \end{itemize} + + \subheading{Enkopplungsregelung von Mehrgrößensystemen} + \textbf{1.} Überprüfen ob Anzahl der Ein- und Ausgangsgrößen $m$ der Regelstrecke gleich ist, sonst funktionierts nicht. \\ + \textbf{2.} Bestimme für jede Ausgangsgröße $v_i(t)$ den relativen Grad, d.h. die kleinste Zahl $r_i$, für die gilt: + \begin{flalign*} + \textbf{C}_i \textbf{A}^0 \textbf{B} &= \textbf{0} \\ + \textbf{C}_i \textbf{A}^1 \textbf{B} &= \textbf{0} \\ + \vdots ~~ &~\vdots \\ + \textbf{C}_i \textbf{A}^{r_i -1} \textbf{B} &\neq \textbf{0}&\\ + \end{flalign*} + \textbf{3.} Bestimme die Matrizen\\ + \begin{flalign*} + \textbf{C}^* + &= + \left[ + \begin{array}{c} + \textbf{C}_1 \textbf{A}^{r_1}\\ + \textbf{C}_2 \textbf{A}^{r_2} \\ + \vdots \\ + \textbf{C}_m \textbf{A}^{r_m} \\ + \end{array} + \right]; ~ + \textbf{D}^* = + \left[ + \begin{array}{c} + \textbf{C}_1 \textbf{A}^{r_1 -1} \textbf{B} \\ + \textbf{C}_2 \textbf{A}^{r_2 -1} \textbf{B} \\ + \vdots \\ + \textbf{C}_m \textbf{A}^{r_m -1} \textbf{B} \\ + \end{array} + \right] &\\ + \end{flalign*} + und überprüfe ob die 'Entkoppelbarkeitsmatrix' \textbf{D}$^*$ ivertierbar ist. Sollte das nicht der Fall sein, so kann das System nicht entkoppelt werden.\\ + \textbf{4.} Durch die Rückführung + \begin{equation*} + \textbf{y}(t) = \textbf{D}^{*^{-1}}(\textbf{w}_r - \textbf{C}^*\textbf{x}) + \end{equation*} + werden die Stellsignale so gewählt, dass $m$ Intergratorenketten der Länge $r_i$ entstehen. Dabei ist der Vektor der benötigten $r_i$-fachen zeitlichen Ableitung der Sollwertsignale + \begin{flalign*} + \textbf{w}_r = + \left[ + \begin{array}{c} + w_1^{(r_1)}\\ + w_2^{(r_2)}\\ + \vdots\\ + w_m^{(r_m)}\\ + \end{array} + \right] = + \left[ + \begin{array}{c} + d^{r_1}w_1/dt^{r_1}\\ + d^{r_2}w_2/dt^{r_2}\\ + \vdots\\ + d^{r_m}w_m/dt^{r_m}\\ + \end{array} + \right] + \end{flalign*} + zu verwenden. \\ + \textbf{5.} Diese entkoppelten Integratorenketten müssen geregelt werden. Dazu kann, muss aber nicht, ein Zustandsregler genutzt werden. Will man eine Festwertregelung anwenden, so gibt man für jede Ausgangsgröße $v_i(t)$ einen Sollwert $w_i(t)$ vor und fordert ein dynamisches Verhalten gemäß der vorgegebenen Führungsübertragungsfunktion + \begin{equation*} + G_{w_i}(s) = \frac{\ell_i}{s^{r_i} + a_{i,r_i-1} s^{r_i-1} + \cdots a_{i,1} s + a_{i,0}} + \end{equation*} + Dies entspricht einer Rückführung der jeweiligen Ausgangsgröße $v_i(t)$ und ihrer ersten $r_i -1$ Ableitungen. Diese erhält man aus dem Zustandsvektor \textbf{x}$(t)$ mit den Beziehungen + \begin{flalign*} + v_i(t) &= \textbf{C}_i\textbf{x}(t)\\ + \dot{v}_i(t) &= \textbf{C}_i\textbf{A}\textbf{x}(t) \\ + \ddot{v}_i(t) &= \textbf{C}_i\textbf{A}^2\textbf{x}(t) \\ + &\vdots + \end{flalign*} + \heading{Trajektorienplanung} + \subheading{Zwei-/dreifach differenzierbare Übergänge} + \textbf{1. Methode: Zwei Parabeläste gleicher Dauer}\\ + Eine erste Lösung erhlät man, wenn man den Übergang in zwei Teile gleicher Länge $t_e/2$ aufteilt und die zweite Ableitung des Sollwerts $\ddot{w}(t)$ in beiden Intervallen konstant wählt. + \ccontent{\includegraphics[width = 0.4\columnwidth]{Fuehrungssignalprozess2Ordnung.png}} + Hinweis: Um zu übersichtlichen Gleichungen zu kommen, wird im Folgenden im $i$-ten Teilintervall mit der 'lokalen Zeit' $\tau_i$ gerechnet, die jeweils vom Beginn des Intervalls zählt. \\ + Im ersten Intervall bis $t = t_e/2$ gilt somit $\tau_1 = t$ und \\ + % \begin{figure}[htbp] + % \begin{minipage}[t]{0.45\columnwidth} + % \vspace{0pt} + % \centering + % \includegraphics[width = \columnwidth]{TrajekMeth1.png} + % \end{minipage} + % \hfill + % \begin{minipage}[t]{0.45\columnwidth} + % \vspace{0pt} + % $\ddot{w}(\tau_1) = a_1$ \\ + % $\dot{w}(\tau_1) = a_1 \cdot \tau_1 + \underbrace{\dot{w}_{10}}_{ = 0} = a_1\cdot \tau_1$ + % \end{minipage} + % \end{figure} + % \begin{flalign*} + % \ddot{w}(\tau_1) &= a_1 \\ + % \dot{w}(\tau_1) &= a_1 \cdot \tau_1 + \underbrace{\dot{w}_{10}}_{ = 0} = a_1\cdot \tau_1 \\ + % w(\tau_1) &= a_1 \cdot \frac{\tau_1^2}{2} + \underbrace{\dot{w}_{10}\tau_1 + w_{10}}_{ = 0} = a_1 \cdot \frac{\tau_1^2}{2} &\\ + % \end{flalign*} + $\ddot{w}(\tau_1) = a_1~~~~~~~~~~~~~~~~$ + $\dot{w}(\tau_1) = a_1 \cdot \tau_1 + \underbrace{\dot{w}_{10}}_{ = 0} = a_1\cdot \tau_1$\\ + $w(\tau_1) = a_1 \cdot \frac{\tau_1^2}{2} + \underbrace{\dot{w}_{10}\tau_1 + w_{10}}_{ = 0} = a_1 \cdot \frac{\tau_1^2}{2}$\\ + % \ccontent{\includegraphics[width = 0.4\columnwidth]{TrajekMeth1.png}} + Im ersten Intervall wird durch die konstante Beschleunigung somt eine linear mit der Zeit steigende Geschwindigkeit aufgebaut. Dabei wird man in den meisten Fällen mit der maximal zulässigen Beschleunigung arbeiten, d.h. $a_1 = a_{Lim}$. \\ + Im zweiten Intervall muss abgebremst werden. Da beide Intervalle gleich lang sind, wird duch $a_2 = -a_{Lim}$ genau am Ende des Bremsvorgangs die GEschwindigkeit null erreicht. Nun ist noch offen, an welchem Zeitpunkt $t_e/2$ umzuschalten ist zwischen $a_1$ und $a_2$ bzw. wie lange (Wert von $t_e$) der gesamte Vorgang dauert. + Am Ende von Intervall 1 ist\\ + $w(\tau_1 = t_e/2) = a_{Lim} \cdot \frac{(t_e/2)^2}{2} = a_{Lim} \cdot \frac{t_e^2}{8}$\\ + und damit gilt am Ende des zweiten Intervalls für den Zusammenhang zwischen dem Endwert $w_e$ und $t_e$\\ + $w_e = a_{Lim} 2 \frac{t_e^2}{8} = a_{Lim}\frac{t_e^2}{4} \rightarrow t_e = \sqrt{\frac{4w_e}{a_{Lim}}}$\\ + \textbf{Beispiel:}\\ + periodischer, überschwingfreier Sollwertwechsel zwischen den Werten $w(t = 1) = 1; w(t=2) = 4 $ zwei Abschnitte mit je 0.5sec. Beschleunigung im ersten Abschnitt ($0 \leq \tau_1 < 0.5$): $a_1 = 1.5\cdot 2/0.5^2 = 12$ erster Abschnitt: $w(t) = w(t=1) + a_1(t-1)^2/2 = 1+12(t-1)^2/2$\\ + Geschwindigkeit am Ende des ersten Abschnitts: $\dot{w}(\tau_1 = 0.5) = a_1*0.5 = 6$. Das Polynom im zweiten Abschnitt ist defniniert duch $w(\tau_2) = w(\tau_2 = 0) + \dot{w}(\tau_2 = 0)\tau_2 + \ddot{w}(\tau_2) \tau_2^2/2$ mit\\ + $w(\tau_2 = 0) = 2.5$ ist der Weg der schon zurück gelegt wurde, bei dieser Methode der halbe Weg $3/2$\\ + $\dot{w}(\tau_2 = 0) = 6$ ist die Geschwindigkeit am Anfang des zweiten Abschnitts bzw. am Ende des ersten.\\ + $\ddot{w}(\tau_2) = -12$\\ + und $\tau_2 = (t-1.5)$ : zweiter Abschnitt: $w(t) = 2.5 + 6(t-1.5) -6(t-1.5)^2$ + \textbf{2. Methode: Zwei Parabeläste gleicher Dauer + Phase mit Konstantgeschwindigkeit}\\ + Nun wird neben der Beschleunigungs- und der Bremsphase ein Abschnitt mit konstanter, maximaler Geschwindigkeit betrachtet. In Phase 1 mit der Dauer $t_1$ und mit $a_1 = a_{Lim}$ beschleunigt, bis $v_{Lim}$ erreicht ist. Dafür ist eine Dauer $t_1 = \frac{v_{Lim}}{a_{Lim}}$ nötig und am Ende dieses ersten Intervalls wurde der Weg $w_1 = \frac{v_{Lim}}{2}t_1$ zurückgelegt. Die Bremsphase 3 dauert wieder genauso lange wie Phase 1 und darin wird der selbe Weg $w_3 = w_1$ zurückgelegt. Schließlich wird die Dauer von Phase 2 bestimmt durh die Endwertbedingung $w_e = w_1 + t_2 v_{Lim} + w_3 \rightarrow t_2 = \frac{w_e - v_{Lim t_1}}{v_{Lim}}$\\ + Bsp: $w(\tau_1) = a_{Lim} \cdot \frac{\tau_1^2}{2};~~ w(\tau_2) = w_1 + v_{Lim}\cdot \tau_2; ~~ w(\tau_3) = w_1+w_2 + v_{Lim}\cdot \tau_3 - a_{Lim}\frac{\tau_3^2}{2}$\\ + Es ergibt sich eine längere Gesamtdauer als bei Methode 1, bei gleichen Werten für $a_{Lim}$ und $v_{Lim}$ aber auch einen weiteren Weg.\\ + \textbf{2a. Methode: Methode 2 mit Interpolationstakt-Quantisierung}\\ + Zunächst wie bei M2 $t_1$ und $t_2$ bestimmen, dann $t_1^* = AUFRUNDEN(t_1)~~~~~t_2^* = AUFRUNDEN(t_2)$. Dadurch verlängerte Phasen, Beschleunigung muss angepasst werden. Reduzierte Geschwindigkeit: $v_{max}^* = a_1^* t_1^*$\\ + \begin{flalign*} + w_e &= \underbrace{v_{max}^* \cdot t_2^*}_{Konstantfahrt} + \underbrace{2\cdot (v_{max}^*/2)t_1^*}_{Beschleunigung + Bremsen} &\\ + a_1^* &= \frac{w_e}{t_1^*(t_2^* + t_1^*)} &\\ + \end{flalign*} + \textbf{3. Methode: mit Ruckbegrenzung}\\ + \begin{flalign*} + t_r &= \frac{a_{lim}}{r_{lim}} ~~~ \text{Zeit für den Beschleunigungsaufbau} &\\ + t_a &= \frac{v_{lim} - a_{lim}^2/r_{lim}}{a_{lim}} ~~~ \text{Zeit für maximale Beschleunigung}&\\ + t_v &= \frac{w_e - v_{lim}(2t_r+t_a)}{v_{lim}} ~~~ \text{Zeit für maximale Geschwindigkeit}&\\ + \end{flalign*} + \ccontent{\includegraphics[width = \columnwidth]{TrajekMeth123.png}} + \heading{Zeitdiskrete Regelung} + \textbf{Zwei Arten:}\\ + \textbf{1. quasikontinuierliche Regelung} bei kleinen Abtastzeiten + \begin{itemize} + \item[a)] Reglerentwurf im Zeitkontinuierlichen $G_R(s)$ + \item[b)] Reglerdiskretisierung $G_(z)$ mit Euler Vortwärtstransormation + \end{itemize} + Faustformel: $T_{ab} \leq \frac{0.1}{\omega_{max}};$ $\omega_{max}$ aus Polbetrachtung: Trage alle Pole (und Nst.) von Strecke, Regler und geschlossenem Regelkreis in Polstellendiagram. Radius des kleinstmöglichen Kreises um Nullpunkt und allen Polen und Nst. entspricht $\omega_{max}$.\\ + \textbf{2. echt zeitdiskrete Regelung} + \begin{itemize} + \item[a)] Strecke diskretisieren $G_s(z)$ mit sprunginvarianter Transformation + \item[b)] Regler zeitdiskret entwerfen $G_R(z)$ $~~~~~~~~~~~T_{ab} \leq \frac{1}{\omega_{max}}$ + \end{itemize} + Hier geht die Tatsache der Abtastung mit in den Reglerentwurf mit ein. + + \subheading{Diskretisierung zeitkontinuierlicher Systeme} + \textbf{Euler-Vorwärtsdifferenz:} $s \approx \frac{z-1}{T_a}$\\ + \textbf{Euler-Rückwärtsdifferenz:} $s \approx \frac{z-1}{zT_a}$\\ + Pro: Einfache Rechnung, Einsatz auch bei nichtlinearen Systemen möglich. Contra: großer Fehler für große Abtastzeiten, Alias Effekte\\ + \textbf{Bilineare Transformation:} $s \approx \frac{2(z-1)}{T_a(z+1)}$\\ + Pro: Genauer als Euler Näherung, zeitdiskrete Realisierung von Filtern. Contra: Näherung des Zeitverhaltens ist zwar besser als bei Euerler, jedoch nicht exakt.\\ + \textbf{Sprunginvariante Transformation:} $z_\infty = e^{s_\infty \cdot T_a}$\\ + \ccontent{\includegraphics[width= \columnwidth]{sprunginvariantenTransformation.png}} + Pro: exakte Lösung für treppenförmige Eigangssignalverläufe (meistes), Zeitdiskrete Frequenzgangsnachbildung ist meist besser als Euler, Diskretisierungsvariante der Wahl wenn zeitdiskrete Bescheibung einer zeitkontinuierlichen Regelstrecke gesucht ist. Contra: lässt sich nicht auf beliebige (nichtlineare) Systeme erweitern + \subheading{Analyse zeitdiskreter Regelungen} + Dynamik eines zeitdiskreten Regelkreises durch \textbf{charakteristische Gleichung}: $1 + H_0(z) = 0$. $H_w = \frac{H_0}{1+H_0}$\\ + Übertragungsfunktion eines offenen Regelkreises bei einschleifigen Regelungen wie im zeitkontinuierlichen: $H_0(z) = H_R(z)H_S(z)$. Zusammen mit der charakteristischen Gleichung entsteht ein von den Polen und der Abtastzeit abhängiger Term. Die Pole sollen zwischen -1 und 1 liegen (am besten aber zwischen 0 und 1 um kein alternierendes Verhalten zu erzeugen) um stabil zu sein, daraus lässt sich ein Bereich für die Abtastzeit besimmen. $T_a \leq~$ kleinste Streckenzeitkonstante (wegen schnellstem Sensor)\\ + \textbf{Grenzwertsätze:}\\ + Anfangswertsatz: $\underset{k\rightarrow 0}{lim}x[k] = \underset{z\rightarrow \infty}{lim} X(z)$ wenn $\underset{k\rightarrow 0}{lim} x[k]$ existiert \\ + Endwertsatz: $\underset{k\rightarrow \infty}{lim}x[k] = \underset{z\rightarrow 1}{lim} (z-1)X(z)$ wenn $\underset{k\rightarrow \infty}{lim} x[k]$ existiert \\ + Für den Sprung-Endwert, Sprung $\frac{z}{z-1}$ nicht vergessen.\\ + \textbf{Beispiel:} Frequenzbereichs-Zustandsreglerentwurf mit I-Anteil und reduziertem Beobachter:\\ + Wegen des unterschiedlichen Endwertsatzes: $N_R(z=1) = 0$ (dann I-Anteil im ZR)\\ + $H_S(z) = \frac{4z + 3}{z^{\textcolor{red}{2}} - 1.7z +0.72}$, alle Pole nach $z_\infty = 0.5$\\ + $n_S = \textcolor{red}{2} \rightarrow \tilde{N}(z) = (z-0.5)^{\textcolor{red}{2}}$\\ + $n_B = \textcolor{red}{2} \underbrace{-1}_{weil reduziert} \underbrace{+1}_{I-Anteil} = \textcolor{Green}{2} \rightarrow \Delta(z) = (z-0.5)^{\textcolor{Green}{2}}$\\ + Entwurfsgleichung:\\ + $\underbrace{(4z+3)}_{Z_S(z)} \underbrace{(b_2z^{\textcolor{Green}{2}} + b_1 z + b_0)}_{Z_V(z)} + \underbrace{(z^2 - 1.7z + 0.72)}_{N_S(z)} \underbrace{(z^{\textcolor{Green}{2}} + a_1 z + a_0)}_{N_R(z)} \overset{!}{=} \underbrace{(z-0.5)^4}_{\Delta(z)\tilde{N}(z)}$\\ + Abschätzung der Einschwingdauer der Regelung (Störverhalten) $\rightarrow$ 4 Pole bei $z_\infty = 0.5 = e^{-T_a/T_x} ~~~ T_x = -T_a/ln(0.5)$ Enschwingdauer $\approx Anz.~Pole \cdot (3...5)\cdot T_x$\\ + Charakteristische Gleichung wie im Zeitkontinuierlichen: $det(z\textbf{I} - \textbf{A} + \textbf{B}\textbf{K}) \overset{!}{=} \tilde{N}(z)$\\ + \textbf{Führungsgrößenfaktor $\ell$:} $\ell = \frac{\tilde{N}(1)}{Z_s(1)} = \frac{1}{\textbf{C}(\textbf{I}-\textbf{A} + \textbf{BK})^{-1}\textbf{B}}$ + + \heading{Zusätzliches} + $ax^2 + bx +c = 0 \rightarrow x_{1/2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$\\ + $(a+b)^3 = a^3 + 3ab^2 + 3a^2b +b^3$ ~~~~~~ $a^m\cdot a^n = a^{m+n};~~ (a^m)^n = a^{m\cdot n}$\\ + System $G(s) =b \frac{as^2}{1+bs+cs^2}$ mithilfe der Sprungantwort bestimmen: \\ + Anfangs- und Entwertsatz (bei Dauerschwingung nur Anfangswertsatz...), wenn Schwinung nicht abklingt also nicht gedämpft ist $\rightarrow b = 0$, Schwinungsperiodendauer: $\omega_0 = \frac{2 \pi}{T} \rightarrow c = \frac{1}{\omega_0^2}$\\ + \textbf{l'Hospital:} Wenn bei Lim 0/0 oder $\infty / \infty$ raus kommt $\rightarrow$ Ableiten. \\ + \textbf{Sprungantwort skizzieren:} Endwertsatz (da den Sprung nicht vergessen $G_w \cdot \frac{1}{s}$), kein Überschwingen wenn reelle Pole und keine Nullstellen + \ccontent{\includegraphics[width= 0.5\columnwidth]{SprungImulsAntwort.png}} + + \textbf{Differenzenordnung r einer Regelstrecke aus Struktur:} durch 'Hinsehen': Weg vom Eingang bis Ausgang mit kleinster Anzahl an Integratoren r = Anzahl der Integratoren. Um Ausnahmen auszuschließen ggf. lieber über $v; \dot{v}; \ddot{v}$ nach $y$ auflösen gehen.\\ + \textbf{Minimalphasigkeit:} System ist minimalphasig wenn Streckennullstellen auf linker s-Halbebene liegen, also kleiner Null sind. Dafür Streckenübertragungsfunktion aus 'Vorwärtsglieder durch (1+Schleifenglieder) + \subheading{Ableitungsregeln} + \begin{flalign*} + (ax^n)^{\prime} &= nax^{n-1}~~~~~(ax(t)^n)^{\prime} = nax(t)^{n-1}\dot{x}(t) &\\ + (\frac{1}{x})^{\prime} &= -\frac{1}{x^2} &\\ + (\sqrt{x})^{\prime} &= \frac{1}{2\sqrt{x}}~~~(\sqrt[3]{x(t)})^{\prime} = (x^{\frac{1}{3}})^{\prime} = \frac{1}{3}x^{-\frac{2}{3}}\dot{x} &\\ + (e^x)^{\prime} &= e^x &\\ + (a^x)^{\prime} &= a^x\cdot ln(a) &\\ + (ln(x))^{\prime} &= \frac{1}{x} &\\ + (sin(x))^{\prime} &= cos(x) &\\ + (cos(x))^{\prime} &= -sin(x) &\\ + (tan(x))^{\prime} &= \frac{1}{(cos(x))^2} &\\ + (f(x) + g(x))^{\prime} &= f^{\prime}(x) + g^{\prime}(x) &\\ + (f(x) \cdot g(x))^{\prime} &= f^{\prime}(x)\cdot g(x) + f(x) \cdot g^{\prime}(x) &\\ + (\frac{f(x)}{g(x)})^{\prime} &= \frac{f^{\prime}(x)\cdot g(x) - f(x) \cdot g^{\prime}(x)}{g(x)^2} &\\ + (f(g(x)))^{\prime} &= f^{\prime}(g(x)) \cdot g^{\prime}(x) &\\ + \end{flalign*} + \subheading{Stabilität und Stabilitätsrand} + Stabilität des Regelkreises: \textit{Aus Skript S.1.10} + \begin{itemize} + \item Liegen alle Pole des Regelkreises in der linken offenen s-Halbebene (d.h. ist der Realteil aller Pole kleiner als Null), so ist der Regelkreis stabil und alle Eigenschwingungen klingen mit der Zeit ab (alle Lösungsantreile aus den Summanden der Partialbruchentwicklung weisen Exponentialfunktionen $e^{a\cdot t}$ bzw. $e^{a\cdot t}\cdot sin(\omega \cdot t + \phi)$ bzw. $t\cdot e^{a\cdot t}\cdot sin(\omega \cdot t + \phi)$ mit $a < 0$ auf). + \item Liegt mindestens ein Pol des Regelkreises in der rechten s-Halbebene, d.h. sein Realteil ist größer als Null, so ist der Regelkreis instabil und mindestens eine aufklingende Eigenschwingung existiert (mindestens ein Lösungsanteil enthält eine Exponentialfunktion $e^{a\cdot t}$ mit $a>0$). + \item Gibt es einfache Pole auf der imaginären Achse, d.h. haben einfache Pole einen verschwindenden Realanteil, so ist der Regelkreis am 'Stabilitätsrand' und kann eine Dauerschwingung ausführen (der zugehörigen Lösungsanteil enthält eine ungedämpfte Schwingung $sin(\omega_x \cdot t + \phi)$). \\ Liegt der einfache Pol im Ursprung, so hat die Dauerschwingung "die Frequezn Null" (der zugehörigen Lösungsanteil $\sigma (t)$ (= Sprungfunktion) ist konstant). + \item Gibt es mindestens einen mehrfachen Pol auf der imaginären Achse, ist der Regelkreis instabil und mindestens eine aufklingende Eigenschwingung existiert. Bei einem doppelten Pol auf der imaginären Achse besipsielsweise enthält der zugehörige Lösungsanteil eine aufklingede Sinus- bzw. Cosinusfunktion $t \cdot sin/(\omega_x \cdot t)$ bzw. $t\cdot cos(\omega_x \cdot t)$) + \end{itemize} + Für Stabilität müssen alle Pole auf der Linken s-Halbebene liegen und auch nicht auf der Im-Achse.\\ + Stabilität eines (linearisierten) Modells: wenn \textbf{A} eine Dreiecksmatrix und alle Hauptdiagonalenelemente = Eigenwerte negativ + % \subheading{ZR mit I-Anteil} + % \ccontent{\includegraphics[width= \columnwidth]{ZustandsR_I_I2_Sin.png}} + \subheading{PI-Regler Einstellen} + \subsubheading{Pol-Nst-Kompensation} + Kann zur Parametrierung eines PI-Reglers verwender werden. Kompensierter (langsame) Zeitkonstante tritt im Führungsverhalten nicht in Erscheinung, im Störverhalten ist der Pol trotzdem sichtbar. Wenn eine sehr langsame Zeitkonstante (1+sT) kompensiert wird kann im Störverhalten kriechendes Verhalten auftreten. Instabile Streckenpole dürfen nicht kompensiert werden. \\ + %\textcolor{magenta}{\textit{Hier evtl noch die anderen Möglichkeiten zum Einstellen von PI-Regler erwähnen (siehe Kapitel 1.11 im Skript)}} + \subsubheading{Betrags- und Symmetrisches Optimum} + \textbf{Betragsoptimum}: Kompensiert langsamste Streckenzeitkonstante, entspricht also Pol-Nst.-Kompensation. Gilt als gute Einstellung fürs Führungsverhalten.\\ + \textbf{Symmetrisches Optimum}: Ohne Kompensation der langsamsten Streckenzeitkonstante. Gilt als gute Einstellung fürs Störverhalten. Kann zu Überschwingern führen.\\ + \ccontent{\includegraphics[height=9cm]{BetragsSymmetrieOptimum.png}} + \subheading{ZR-Berechnung mit Matlab} + acker: + \begin{itemize} + \item für Eingößensysteme + \item beliebige Polverteilung + \item kann bei großer Systemordnung numerisch kritisch werden + \end{itemize} + place: + \begin{itemize} + \item auch für Mehrgrößensysteme anwendbar + \item numerisch robuster + \item Einschränkung bei mehrfacher Polvorgabe + \end{itemize} + +\end{multicols*} + +\begin{multicols*}{2} +\subheading{Sinusförmige Führungssignale} +\textit{Zusätzlich zu Seite 4}\\ +\ccontent{\includegraphics[height=3.5cm]{SinusStoermodell.pdf}} +\begin{flalign*} + \left[ + \begin{array}{c} + \dot{\textbf{x}}(t) \\ + \dot{x}_{sin,1}(t) \\ + \dot{x}_{sin,2}(t) \\ + \end{array} + \right] + &= + \underbrace{ + \left[ + \begin{array}{ccc} + \textbf{A} & 0 &0\\ + \textbf{0}&0&1\\ + \textbf{C} & -\omega_0^2 &0\\ + \end{array} + \right]}_{\textbf{A}_{sin}} + \left[ + \begin{array}{c} + \textbf{x}(t) \\ + x_{sin,1}(t) \\ + x_{sin,2}(t) \\ + \end{array} + \right] + + \underbrace{ + \left[ + \begin{array}{c} + \textbf{B} \\ + 0 \\ + 0\\ + \end{array} + \right]}_{\textbf{B}_{sin}} + y(t) + + \left[ + \begin{array}{c} + \textbf{0} \\ + 0\\ + -1 \\ + \end{array} + \right] w(t) & +\end{flalign*} +\textbf{C}$_{sin} = [\textbf{C}~0~0]; \textbf{K}_{sin} = [k_1~k_2~k_3~k_{sin,1}~k_{sin,2}]$\\ +\subheading{System mit Beobachter, kompensierender Rückführung und Zustandsregler mit I-Erweiterung} +\ccontent{\includegraphics[width= 0.95\columnwidth]{SS15A1_9.png}} +\subheading{Überschwinger berechnen} +\begin{flalign*} + \text{Dämpfung}~D = -\frac{ln("u)}{\sqrt{\pi^2 + ln("u)^2}}~~~ \text{Mit ü = 0.05 für 5\% Überschwinger}\\ +\end{flalign*} +PT$_2$-Anteil: $s^2 + 2D\omega_0s + \omega_0^2$, Beitrag zur Summenzeitkonstante: $2D/\omega_0$ +\end{multicols*} + +\end{document} \ No newline at end of file diff --git a/README.md b/README.md new file mode 100644 index 0000000..0457628 --- /dev/null +++ b/README.md @@ -0,0 +1,3 @@ +Fehler bitte sofort melden! Änderungsvorschläge auch gerne melden. + +Formelsammlung für ENT4.