From 78366caf2336d3466ef17e4f604fcfc1516ba15c Mon Sep 17 00:00:00 2001
From: Jan Falkenhain <jan.falkenhain@hs-bochum.de>
Date: Thu, 1 Apr 2021 19:58:02 +0200
Subject: [PATCH] 5rd commit

---
 1_Einleitung.tex                | 109 +++++++++++++-------
 2_Grundlagen.tex                | 176 ++++++++++++++++++++++++++------
 3_Euler-Diskretisierung.tex     |   6 +-
 4_ZOH-Diskretisierung.tex       |  13 +--
 5_Impulse-Diskretisierung.tex   |   2 +-
 6_FOH-Diskretisierung.tex       |  41 +++++++-
 7_Tustin-Diskretisierung.tex    |  50 ++++++++-
 8_Matched-Filter.tex            |   4 +-
 FB_Diskretisierung.pdf          | Bin 989080 -> 1034021 bytes
 Matlab/Diskretisierung_FOH.m    |  48 +++++++++
 Matlab/Diskretisierung_Tustin.m |  44 ++++++++
 anhang.tex                      |  34 +++++-
 deckblatt-cvh.tex               |   4 +-
 hauptdatei.bib                  |  17 +++
 hauptdatei.tex                  |  20 +++-
 15 files changed, 474 insertions(+), 94 deletions(-)
 create mode 100644 Matlab/Diskretisierung_FOH.m
 create mode 100644 Matlab/Diskretisierung_Tustin.m

diff --git a/1_Einleitung.tex b/1_Einleitung.tex
index 150d664..8977ac8 100644
--- a/1_Einleitung.tex
+++ b/1_Einleitung.tex
@@ -1,24 +1,49 @@
 \section{Einleitung%
          \label{sec:Einleitung}}
 
-Diskrete Systeme werden im Rahmen der digitalen Signalverarbeitung und der Regelungstechnik entworfen, in letzterer zumeist mit Bezug zu kontiuierlichen, physikalischen Modellen. Zwei wesentliche Motive können hier vorliegen, um kontinuierliche System zu diskretisieren, also sie durch ein mit fester Abtastzeit arbeitendes, diskretes System zu approximieren.
+Diskrete Systeme werden im Rahmen der digitalen Signalverarbeitung und der Regelungstechnik entworfen, in letzterer zumeist mit Bezug zu kontiuierlichen, physikalischen Modellen. Zwei wesentliche Motive können hier vorliegen, um kontinuierliche Systeme zu diskretisieren, also sie durch ein mit fester Abtastzeit arbeitendes, diskretes System zu approximieren.
 
 \begin{enumerate}
-\item Hat man mit klassischen Methoden einen kontinuierlichen Regler entworfen, z. B. anhand eines Streckenmodells, dann erfordert die Realisierung auf einem Rechner (einer echtzeitfähigen Hardware beliebiger Art) eine diskrete Approximation, also ein unter Beachtung der Abtastzeit \textbf{den kontinuierlichen Regler} möglichst gut nachbildendes diskretes Modell. Dies ist immer erforderlich, wenn nicht Bibliotheken für die automatisierte Übersetzung kontinuierlicher Parameter, z. B. für PID-Regler, oder andere Software zur Verfügung steht, welche eine numerische Lösung bieten (wie z. B. MATLAB Coder).  
+\item Hat man mit klassischen Methoden einen kontinuierlichen Regler entworfen, z. B. anhand eines Streckenmodells, dann erfordert die Realisierung auf einem Rechner (einer echtzeitfähigen Hardware beliebiger Art) eine diskrete Approximation, also ein unter Beachtung der Abtastzeit \textbf{den kontinuierlichen Regler} möglichst gut nachbildendes diskretes Modell. Dies ist immer erforderlich, wenn nicht Bibliotheken für die automatisierte Übersetzung kontinuierlicher Parameter, z. B. für PID-Regler, oder andere Software zur Verfügung steht, welche eine numerische Lösung bieten (wie z. B. MATLAB\texttrademark-Coder).  
 \item Eine etwas andere Vorgehensweise des Reglerentwurfs besteht darin, den diskreten Regler direkt anhand eines diskreten Streckenmodells zu entwerfen. Dafür ist jetzt also die \textbf{Strecke bzw. das kontinuierliche Streckenmodell} zu diskretisieren. Anschließend steht eine eigene Welt von Methoden für den diskreten Reglerentwurf zur Verfügung, welche oft Ähnlichkeiten mit klassischen kontinuierlichen Methoden aufweisen, aber letztlich eigenständige Methoden darstellen.
 \end{enumerate}
 
-Der Verdacht liegt nahe, dass man in beiden Fällen anstrebt, das kontinuierliche System möglichst gut zu approximieren. Allerdings gilt für den zweiten Fall der Streckendiskretisierung, dass man aus Sicht des zu entwerfenden Reglers tatsächlich das Verhalten des kontinuierliche Modells in Reihe geschaltet mit zwei Haltegliedern erster Ordnung nachbilden will. Dies erzwingt die Eigenschaft des diskreten Systems, pro Abtastzeit nur einen Wert zu verarbeiten und am Ausgang einen Wert für eine Abtastzeit zu halten.
+Der Verdacht liegt nahe, dass man in beiden Fällen anstrebt, das kontinuierliche System \textit{möglichst gut} zu approximieren. Allerdings gilt für den zweiten Fall der Streckendiskretisierung, dass man aus Sicht des zu entwerfenden Reglers tatsächlich das Verhalten des kontinuierliche Modells in Reihe geschaltet mit einem Halteglied erster Ordnung (ZOH, Zero-Order-Hold) nachbilden will. Das resultierende System heißt Abtastsystem, worauf in Kapitel 3 eingegangen wird. Man erhält es durch ZOH-Diskretisierung \ref{sec:ZOH-Diskretisierung} der kontinuierlichen Strecke. Das so erzeugte diskrete System liefert Werte am Ausgang, welche exakt auf der kontinuierlichen Antwort des Streckenmodells liegen, wenn dieses durch ein Halteglied erster Ordnung (ZOH) erzeugte, mit der Periode der Abtastzeit stufenförmige Eingangssignale erhält. Diese Forderung der Exaktheit aus Perspektive des Regler unterscheidet sich fundamental von der Aufgabe, den Regler möglichst gut zu approximieren. Dazu werden in der Praxis die Tustin-Diskretisierung \ref{sec:Tustin-Approximation} oder FOH-Diskretisierung \ref{sec:FOH-Diskretisierung} verwendet, welche auf "`schnellere"' diskrete Systeme führen. 
 
-Einige wichtige Begriffe der Systemtheorie werden im Grundlagenkapitel eingeführt. Dann wird die Euler-Approximation behandelt (Kap \ref{sec:Euler-Diskretisierung}), bei der Ableitungen durch einen simplen Differenzenquotienten ersetzt werden.
+In der digitalen Signalverarbeitung approximiert man kontinuierliche Filter u. a. aus den selben Motiven wie in der Regelungstechnik. Die klassischen Entwurfsverfahren im kontinuierlichen können so eingesetzt werden, bevor zum Schluss die Transformation erfolgt. Andererseits wurden in beiden Disziplinen Methoden für den direkten Systementwurf diskreter Systeme aus dafür formulierten Anforderungen entwickelt. Die Approximation kontinuierlicher Systeme zielt typischerweise je nach Anwendung auf unterschiedliche Eigenschaften, wie den Frequenzgang oder die Impulsantwort bei der Impuls-Diskretisierung \ref{} ab. Auch der Matched-Filter Ansatz \ref{}, welche Pole und Nullstellen des diskreten Systems gezielt setzt, spielt in der digitalen Signalverarbeitung eine größere Rolle.
 
- Mit MATLAB\texttrademark (R2021a) lassen sich Systeme erzeugen, Darstellungen ineinander überführen und auch diskrete Systeme ableiten, wie das  folgende Skript zeigt:\\
+Weitere Motive zur Ermittlung diskreter Systeme können in der Analyse, Simulation, oder modellbasierten Diagnose technischer Systeme liege. Di Ermittlung eines Modells aus Messungen an einem realen System wird als Identifikation bezeichnet. Ein diskretes Modell ist oft der einfachere Weg, so dass bei Bedarf in einem zweiten Schritt in ein kontinuierliches Modell transformiert werden kann. Besonders einfach sind die Ansätze der Euler-Diskretisierung \ref{sec:Euler-Diskretisierung} welche schnell auch auf nicht-lineare Modelle angewendet werden können. Nach Rekapitulation einiger wichtige Begriffe der Systemtheorie im Grundlagenkapitel \ref{} sollen in Kapitel \ref{2} alle genannten Ansätze kurz, aber auch aus verschiedenen Blickwinkeln betrachtet werden, um einen systematischen Überblick zu ermöglichen.
 
-\href{https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/Diskretisierung_mit_MATLAB.m}{Matlab\texttrademark(R2021a) file on Git: Diskretisierung\_mit\_MATLAB.m}
+\begin{landscape}
 
-%\url{ https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/Diskretisierung_mit_MATLAB.m}
+\begin{table}[H]
+  \caption{Diskretisierungsansätz und deren Eigenschaften%
+           \label{table:massen}}
+  %\medskip
+  \begin{center}
+    \begin{tabular}{l|c|c|c|c|c}
+      Verfahren                 & $s\mapsto z$ ? &  Differenziator&Integrator& Zeitfunktion& Pol-Trafo\\
+      \hline
+      Forward-Euler             & Ja 		&  $(z-1)/T$&$T\frac{1}{z-1}$ & &$z_i =s_iT + 1$\\
+      Backward-Euler            & Ja  	&  $\frac{z-1}{zT}$&$\frac{Tz}{z-1}$ & &$z_i = \frac{1}{1-s_iT}$\\
+      ZOH   										& Nein	&  & &    h(kT)=h(t)&$z_i=e^{s_i T}$ \\
+      FOH       								&	Nein			&-	 & & &$z_i=e^{s_i T}$\\
+      Impuls                  	& Nein      &&$\frac{z}{z-1}$	 &g(kT)=g(t)&$z_i=e^{s_i T}$\\
+      Tustin                    & Ja       && $\frac{T}{2} \frac{z+1}{z-1}$  & -&\\
+      Mathched Filter 					& Nein   & -&-&  &$z_i=e^{s_i T}$  \\
+      %Holzplatte               &       14 \\
+      %Drehzahlsensoren         &        5 \\
+      \hline
+      %Gesamtmasse              &      226 \\
+    \end{tabular}
+  \end{center}
+\end{table}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{landscape}
+%% Quertabelle Ende
 
-Hier wird ein System zweiter Ordnung (ohne Durchgriff, mit komplexen Eigenwerten) diskretisiert nach den Methoden ZOH (Zero-Order Hold), FOH (First-Order Hold), Impulse und Tustin, welche in den folgenden Kapiteln behandelt werden. Bild \ref{fig:Sprungantwort} vergleicht die Sprungantworten der diskreten Systeme mit der des Originalsystems.
+Die Ansätze können, für die hier behandelten lineare Systeme, sowohl im Zeit- als auch im Frequenzbereich betrachtet werden. Einige Methoden lassen sich durch Ersetzen aller Integratoren in der jeweiligen kontinuierlichen Darstellung der Differenzialgleichung durch diskrete Akkumulatoren umsetzten. Generell ist von Interesse, wie letztlich jede methode die Integration im diskreten numerisch umsetzt. Ob ein stabiles kontinuierliches System in ein stabiles diskretes transforert wird, hängt von der Abbildung der Polstellen ab, für dynamische Eigenschaften sind aber Nullstellen relevant. Diese können abgebildet werden oder neu bei der Transformation entstehen. Wichtige Eigenschaften werden in Tabelle \ref{table:massen} aufgeführt. Ihre Bedeutung soll aber in den einzelnen Kapitel näher erläuter werden. Neben der Übertragungsfunktion als Transformationspaar sollen auch Zustandsraummodelle betrachtet werden.  
 
 \begin{figure}[H]
   \begin{center}
@@ -28,7 +53,13 @@ Hier wird ein System zweiter Ordnung (ohne Durchgriff, mit komplexen Eigenwerten
   \end{center}
 \end{figure}
 
-Der Verlauf für das ZOH-approximierte System liegt zu den Zeitpunkten $kT$ exakt auf der kontinuierlichen Sprungantwort, da dieses Verfahren gerade so konzipert ist. Tustin- und FOH-approximiertes System liefern eine ähnliche Sprungantwort, ohne exakt die kontinuierliche Kurve zu treffen. Da sie die gleiche statische Verstärkung (hier $k_{DC}=1$) wie das Originalsystem haben, verschwinden die Abweichungen nach Abklingen des Übergang. Das dies für die Impulse-approximierte Sprungantwort nicht gilt, kann zu dem Schluss führen, dass Impulse-Approximation wenig geeignet ist, das kontinuierliche System zu approximieren. 
+Für lineare System sind entsprechende Algorithmen für beide Beschreibungsformen in MATLAB\texttrademark (R2021a) größtenteils implementiert.
+Damit lassen sich Systeme erzeugen, Darstellungen ineinander überführen und diskrete Systeme ableiten, wie das  folgende Skript zeigt:
+
+\href{https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/Diskretisierung_mit_MATLAB.m}{MATLAB\texttrademark(R2021a) file on Git: Diskretisierung\_mit\_MATLAB.m}
+
+Hier wird ein System zweiter Ordnung (ohne Durchgriff, mit komplexen Eigenwerten) diskretisiert nach den Methoden ZOH (Zero-Order Hold), FOH (First-Order Hold), Impulse und Tustin. Bild \ref{fig:Sprungantwort} vergleicht die Sprungantworten der diskreten Systeme mit der des Originalsystems.
+Der Verlauf für das ZOH-approximierte System liegt zu den Zeitpunkten $kT$ exakt auf der kontinuierlichen Sprungantwort, da dieses Verfahren gerade so konzipert ist. Tustin- und FOH-approximiertes System liefern eine ähnliche Sprungantwort, ohne exakt die kontinuierliche Kurve zu treffen. Da sie die gleiche statische Verstärkung (hier $k_{DC}=1$) wie das Originalsystem haben, verschwinden die Abweichungen nach Abklingen des Übergang. Dass dies für die Impulse-approximierte Sprungantwort nicht gilt, kann zu dem Schluss führen, dass Impulse-Approximation wenig geeignet ist, das kontinuierliche System zu approximieren. 
 
 \begin{figure}[H]
   \begin{center}
@@ -38,40 +69,40 @@ Der Verlauf für das ZOH-approximierte System liegt zu den Zeitpunkten $kT$ exak
   \end{center}
 \end{figure}
 
-Aus einer anderen Perspektive wird aber lediglich ein anderer Aspekt, nämlich die Impulsantwort möglichst gut approximiert, was dazu führt, dass die Sprungantwort weniger gut getroffen wird. 
+Aus einer anderen Perspektive wird aber lediglich ein anderer Aspekt, nämlich die Impulsantwort möglichst gut bzw. hier approximiert. Ein Vergleich der diskreten mit der kontinuierliche Impulsantwort zeigt Bild \ref{fig:Impulseantwort} 
 
 
 
 
-\begin{landscape}
-
-
-\begin{table}[H]
-  \caption{Diskretisierungsansätz und deren Eigenschaften%
-           \label{table:massen}}
-  \medskip
-  \begin{center}
-    \begin{tabular}{l|c|c|c|c}
-      Verfahren                 & $s\mapsto z$ ? & Integrator& Differenziator& Zeitfunktion\\
-      \hline
-      Forward-Euler             & Ja 		&  & &\\
-      Backward-Euler            & Ja  	&  & &\\
-      ZOH   										& Nein	&  & &    5 \\
-      FOH       								&	Nein			&	 & &20 \\
-      Impuls                  	& Nein      &$\frac{z}{z-1}$	& &g(kT)=g(t)\\
-      Tustin                    & Ja      10 &&&\\
-      Mathched Filter 					& Nein   & -&-&   80 \\
-      %Holzplatte               &       14 \\
-      %Drehzahlsensoren         &        5 \\
-      \hline
-      %Gesamtmasse              &      226 \\
-    \end{tabular}
-  \end{center}
-\end{table}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\end{landscape}
-%% Quertabelle Ende
+%\begin{landscape}
+%
+%
+%\begin{table}[H]
+  %\caption{Diskretisierungsansätz und deren Eigenschaften%
+           %\label{table:massen}}
+  %\medskip
+  %\begin{center}
+    %\begin{tabular}{l|c|c|c|c|c}
+      %Verfahren                 & $s\mapsto z$ ? &  Differenziator&Integrator& Zeitfunktion& Pol-Trafo\\
+      %\hline
+      %Forward-Euler             & Ja 		&  $(z-1)/T$&$T\frac{1}{z-1}$ & &$z_i =s_iT + 1$\\
+      %Backward-Euler            & Ja  	&  $\frac{z-1}{zT}$&$\frac{Tz}{z-1}$ & &$z_i = \frac{1}{1-s_iT}$\\
+      %ZOH   										& Nein	&  & &    h(kT)=h(t)&$z_i=e^{s_i T}$ \\
+      %FOH       								&	Nein			&-	 & & &$z_i=e^{s_i T}$\\
+      %Impuls                  	& Nein      &&$\frac{z}{z-1}$	 &g(kT)=g(t)&$z_i=e^{s_i T}$\\
+      %Tustin                    & Ja       && $\frac{T}{2} \frac{z+1}{z-1}$  & -&\\
+      %Mathched Filter 					& Nein   & -&-&  &$z_i=e^{s_i T}$  \\
+      %%Holzplatte               &       14 \\
+      %%Drehzahlsensoren         &        5 \\
+      %\hline
+      %%Gesamtmasse              &      226 \\
+    %\end{tabular}
+  %\end{center}
+%\end{table}
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\end{landscape}
+%%% Quertabelle Ende
 
 
 %\input{prefix}
diff --git a/2_Grundlagen.tex b/2_Grundlagen.tex
index 4af127f..0b8f45b 100644
--- a/2_Grundlagen.tex
+++ b/2_Grundlagen.tex
@@ -1,7 +1,17 @@
-\section{Grundlagen der Systemtheorie -- eine Auswahl%
+\section{Grundlagen%
          \label{sec:Grundlagen}}
-Betrachtet werden lineare, zeitinvariante Systeme, deren Übertragungsverhalten aufgrund der Linearität im Frequenzbereich durch die Übertragungsfunktion $G(s)$ beschrieben werden kann. Im Zeitbereich wird das System vollständig durch charakteristische Funktionen wie die Impulsantwort $g(t)$ beschrieben, welches mit der Übertragungsfunktion ein Laplace-Transformationspaar bildet:
-\[g(t)~~\laplace~~ G(s)=\frac{Y(s)}{U(s)} = \frac{\Pi_{i=1}^q (s-s_{0i})}{\Pi_{i=1}^n (s-s_i)}\]
+				
+				
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Systemtheorie -- eine Auswahl%
+         \label{sec:GrundlagenST}}	
+							
+\textbf{Zeit- und Frequenzbereich, kontinuierlich und diskret.} Betrachtet werden lineare, zeitinvariante Systeme, deren Übertragungsverhalten aufgrund der Linearität im Frequenzbereich durch die Übertragungsfunktion $G(s)$ beschrieben werden kann. Im Zeitbereich wird das System vollständig durch charakteristische Funktionen wie die Impulsantwort $g(t)$ beschrieben, welche mit der Übertragungsfunktion ein Laplace-Transformationspaar bildet:
+\[g(t)~~\laplace~~ 
+%G(s)
+%=\frac{Y(s)}{U(s)} = \frac{\Pi_{i=1}^q (s-s_{0i})}{\Pi_{i=1}^n (s-s_i)}
+G(s) = \int_0^\infty g(t)e^{-st} dt =L\{g(t)\}
+\]
 Mit der Impulsantwort reagiert das System auf Anregung mit dem Dirac-Impulse
 \begin{align}
 \rho(t)=
@@ -15,20 +25,73 @@ Mit der Impulsantwort reagiert das System auf Anregung mit dem Dirac-Impulse
 \left\{
 \begin{array}{l}
 0~~\text{für}~t_0<0 \\
-1~~\text{für}~t_0\geq0 \neq 0\\
+1~~\text{für}~t_0\geq0 \\
 \end{array}
 \right. 
-=\sigma(t_0)
+=\sigma(t_0)~~.
 \end{align}
+Integration über den Dirac-Impuls liefert die Sprungfunktion $\sigma(t)$, auf die das System mit der Sprungantwort 
+\[h(t)= \int_0^{\infty}g(\tau)d\tau\]
+reagiert. Für stabile Systeme klingt die Impulsantwort ab und die Sprungantwort ist beschränkt.\\
+
+\begin{tikzpicture}[node distance=2.5cm,auto,>=latex', scale=3]
+    \node [int] (a) {$G(s)$};   %\node [int, pin={[init]above:$v_0$}] (a) {$\int$};
+    \node (b) [left of=a,node distance=2cm, coordinate] {a};
+    \node (c) [right of=a] {$$}; %\node [int, pin={[init]above:$p_0$}] (c) [right of=a] {$\int$};
+    \node (d) [right of=c] {$$};
+    \node (e) [right of=d] {$u(t)\laplace U(s)$;~~$y(t)\laplace Y(s)$;~~};
+    \node [coordinate] (end) [right of=c, node distance=2cm]{};
+    \path[->] (b) edge node {$u(t)$} (a);
+    \path[->] (a) edge node {$y(t)$} (c);
+\end{tikzpicture}
 
-Das Ausgangssignal $y(t)$ ergibt sich durch das Faltungsintegral des Eingangs $u(t)$ mit der Gewichtsfunktion
+Das Ausgangssignal $y(t)$ ergibt sich durch das Faltungsintegral der Gewichtsfunktion mit einem beliebigen Eingang $u(t)$ nach
+\[
+y(t) = u(t)\ast g(t) = \int_{t}^t u(\tau) g(t-\tau) d\tau~~.
+\]
+Die Übertragungsfuktion als Bruch zweier Polynome in $s$
 \[
-y(t) = u(t)\ast g(t) = \int_{-\infty}^t u(\tau) g(t-\tau) d\tau~~.
+G(s) = \frac{Y(s)}{U(s)} = \frac{\Pi_{i=1}^q (s-s_{0i})}{\Pi_{i=1}^n (s-s_i)}
 \]
+kann u. a. in Pol-/Nullstellenform oder durch Partialbruchzerlegung in Summanden zerlegt werden.\\
+
 
-Integration über den Dirac-Impuls liefert die Sprungfunktion, auf die das System mit der Sprungantwort $h(t)$ reagiert. Für stabile Systeme klingt die Impulsantwort ab und die Sprungantwort ist beschränkt.\\
+Abtastung wird mathematisch zunächst im Rahmen kontinuierlicher Signale als Gewichtung des Eingangssignals mit einer Dirac-Impulsfolge beschrieben.
+
+\begin{tikzpicture}[node distance=2.5cm,auto,>=latex', scale=3]
+    \node [int] (a) {Abtaster};   %\node [int, pin={[init]above:$v_0$}] (a) {$\int$};
+    \node (b) [left of=a,node distance=2cm, coordinate] {a};
+    \node (c) [right of=a] {$$}; %\node [int, pin={[init]above:$p_0$}] (c) [right of=a] {$\int$};
+    \node (d) [right of=c] {$$};
+    \node (e) [right of=d] {$y^*(t)=u(t) \cdot \sum_{-\infty}^\infty \rho(t-kT)$};
+    \node [coordinate] (end) [right of=c, node distance=2cm]{};
+    \path[->] (b) edge node {$u(t)$} (a);
+    \path[->] (a) edge node {$y^*(t)$} (c);
+\end{tikzpicture}
 
-Durch Abtastung beschränkter kontinuierlicher Signale mit der Abtastzeit $T$ erhält man Folgen, wie beispielsweise die Sprungfolge $\sigma(k)$ für $k\in \mathds{Z}$ aus der Sprungfunktion. Zu beachten ist, dass der Einheitsimpuls
+Durch Abtastung beschränkter, kontinuierlicher Signale mit der Abtastzeit $T$ ergibt sich die Darstellung
+\[
+y^*(t)=\sum_{-\infty}^\infty \rho(t-kT) \underbrace{u(kT)}_{u_k(kT) =u_d(k)}
+\]
+welche den Übergang zur Folge $u_d(k)$ nahelegt. Dieser Übergang wird unter dem Begriff der Abtastung oft verstanden. 
+Beispielsweise resultiert Sprungfolge $\sigma(k)$ für $k\in \mathds{Z}$ aus Abtastung der Sprungfunktion
+\[
+\sigma_d(k) = \sigma_k(kT)~~,
+\] 
+wobei die Indizes $d, k$ für diskretes und kontinuierliches System nur bei Verwechselungsgefahr gesetzt werden.
+Die Laplace-Transformation des abgetasteten pseudo-kontinuierlichen Signals
+\begin{align}
+L\{y^*(t)\}&=
+\int_0^\infty \sum_{-\infty}^\infty \rho(t-kT) u(kT)e^{-st} dt\\
+=&\sum_{0}^\infty u(kT) \int_0^\infty \rho(t-kT) e^{-st} dt\\
+=&\sum_{0}^\infty u(kT)  e^{-skT}
+\end{align}
+motiviert den Übergang zur z-Transformation durch ersetzen von $z=e^{-sT}$ für diskrete Folgen $u_d(k)$
+\begin{align}
+Z\{u(k)\}=\sum_{0}^\infty u(k)  z^{k} = U(z)
+\end{align}
+
+Zu beachten ist, dass der Einheitsimpuls
 \begin{align}
 \rho(k)=
 \left\{
@@ -38,17 +101,23 @@ Durch Abtastung beschränkter kontinuierlicher Signale mit der Abtastzeit $T$ er
 \end{array}
 \right. =\sigma(k)-\sigma(k+1)~~
 \end{align}
-nicht aus der Abtastung des Dirac-Impulses hervorgeht, da dieser bei null nicht beschränkt ist. Die Bedeutung des diskreten Impulses ist also eine völlig andere, nämlich dass alle diskreten Folgen, im weiteren auch Signale genannt, eine Summe von gewichteten Impulsen darstellen, verschoben um den Zeitindex $k$.
+nicht aus der Abtastung des Dirac-Impulses hervorgeht, da dieser bei null nicht beschränkt ist. Die Bedeutung des diskreten Impulses ist also eine etwas andere, nämlich dass alle diskreten Folgen eine Summe von gewichteten Impulsen darstellen, verschoben um den Zeitindex $k$.
 
-Ersetz man in der Laplace-Transformation $z=e^{-sT}$, so entsteht die z-Transformation, über welche die diskrete Impulsantwort und die diskrete Übertragungsfunktion ein z-Transformationspaar bilden
-\[g(k)~~\laplace~~ G(z) = \frac{Y(z)}{U(z)} = \frac{\Pi_{i=1}^q (z-z_{0i})}{\Pi_{i=1}^n (z-z_i)}\]
-und sich ähnliche Zusammenhänge wie im Kontinuierlichen ergeben (siehe Seite \ref{ref:pdfuebersicht}).
+%Ersetz man in der Laplace-Transformation , so entsteht die z-Transformation, über welche 
+Die diskrete Impulsantwort als Reaktion diskreten Impuls $\rho(k)$ und die diskrete Übertragungsfunktion bilden ein z-Transformationspaar 
+\[g(k)~~\laplace~~ G(z) = \frac{Y(z)}{U(z)} = \frac{\Pi_{i=1}^q (z-z_{0i})}{\Pi_{i=1}^n (z-z_i)}~~,\]
+womit sich ähnliche Darstellungen und Zusammenhänge wie im Kontinuierlichen ergeben (siehe nächste Seite 
+%\pageref{ref:pdfuebersicht} funktionieren nicht
+%\ref{ref:pdfuebersicht}
+).
 Über die diskrete Faltung 
 \[
 y(k) = u(k)\ast g(k) = \sum_{j=-\infty}^k u(j) g(k-j)
 \]
 ergibt sich die Systemantwort im Zeitbereich.
-\includepdf[pages=-]{bilder/Systemantworten}\label{ref:pdfuebersicht}
+
+%\label{ref:pdfuebersicht}
+\includepdf[pages=-]{bilder/Systemantworten}
 
 Die Abbildung $z=e^{-sT}$ aus der $s$-Ebene in die $z$-Ebene ist in Bild \ref{fig:s-z} dargestellt. Tatsächlich wird die linke Halbebene in den Einheitskreis abgebildet und die Pole stabiler, diskreter Systeme müssen sich auch in diesem befinden.
 
@@ -70,36 +139,83 @@ Die Abbildung $z=e^{-sT}$ aus der $s$-Ebene in die $z$-Ebene ist in Bild \ref{fi
 \end{center}
 \end{figure}
 
-Im Zeitbereich entstehen Modelle aus Differenzialgleichungen, welche im linearen Fall zumeist in Form eines Zustandsraummodells (ZRM) dargestellt werden können:
+\textbf{Lineares Zusatndsraummodell.} Im Zeitbereich entstehen Modelle aus Differenzialgleichungen, welche zumeist in Form eines Zustandsraummodells (ZRM) dargestellt werden können, im linearen Fall:
 \begin{align}\label{ZRM_kont}
  \dot{\jv x}(t) &= \jv A_k \jv x(t) + \jv B_k \jv u(t),~~\jv x(0)=\jv x_0\\
 \jv y(t) &= \jv C_k \jv x(t) + \jv D_k \jv u(t)\nonumber
 \end{align}
-Darin bezeichnet $\jv x$ den Zustandsvektor. Über die Matrix $\jv A_k$ wird ein System von Differenzialgleichungen erster Ordnung formuliert. $\jv D_k$ wird als Durchgriff bezeichnet, da es die direkte Wirkung eines Eingangssignals $\jv u(t)$ (z. B. P-Anteil beim Regler) auf den Ausgang ohne Integration beschreibt. 
+Darin bezeichnet $\jv x$ den Zustandsvektor. Über die Systemmatrix $\jv A_k$ wird ein System von Differenzialgleichungen erster Ordnung formuliert. $\jv D_k$ wird als Durchgriff bezeichnet, da es die direkte Wirkung eines Eingangsvektors $\jv u(t)$ (z. B. P-Anteil beim Regler) auf den Ausgangsvektor $\jv y(t)$ ohne Integration beschreibt. 
 Die Laplace-Transformierte der Zustandsgleichung lautet
 \begin{align*}
- s{\jv X}(s) &= \jv A_k \jv X(s) + \jv B_k \jv U(s)~~.
+ s{\jv X}(s) &= \jv A \jv X(s) + \jv B \jv U(s)~~,
 \end{align*}
-Aufgelöst nach $\jv X(s)$ und eingesetzt in die Ausganbegleichung ermöglicht die Angabe der Übertragungsfunktion in Abhängigkeit von den Matrizen des ZRMs
+wobei der Index $k$ wegen Eindeutigkeit entfällt. 
+Aufgelöst nach $\jv X(s)$ und eingesetzt in die transformiert Ausgabegleichung ergibt sich die Übertragungsfunktion in Abhängigkeit von den Matrizen des ZRMs
 \begin{align*}
- {\jv G}(s) &= \jv C(s\jv I - A_k)^{-1} \jv B_k +\jv D;~~\jv K_s=\jv G(0) = -\jv C \jv A^{-1} \jv B +D 
+ {\jv G}(s) &= \jv C(s\jv I - \jv A)^{-1} \jv B +\jv D;~~~~\jv K_s=\jv G(0) = -\jv C \jv A^{-1} \jv B +\jv D 
 \end{align*}
-Für $s=0$ ergibt sich die statische Verstärkung.
+Für $s=0$ (bzw. $\dot{\jv x}(t)=0$) ergibt sich die statische Verstärkung $K_s$.
 
-Analog dazu ist
+Analog dazu ist mit dem platzsparenderem Zeitindex ($u(k)=u_k$)
 \begin{align}\label{ZRM_disk}
-\jv x(k+1)&=\jv A_d \jv x(k)+\jv B_d \jv u(k),~~\jv x(0)=\jv x_0\\
-\jv y(k) &= \jv C_d \jv x(k) + \jv D_d \jv u(k)\nonumber
+\jv x_{k+1}&=\jv A_d \jv x_k+\jv B_d \jv u_k,~~\jv x_0\\
+\jv y_k &= \jv C_d \jv x_k + \jv D_d \jv u_k\nonumber
 \end{align}   
-ein diskretes ZRM mit Zustandsübergangs- und Ausgabegleichung . Die Auswertung erfordert immer die Kenntnis des Anfangszustands $\jv x_0$. Die hier verwendete Schreibweise wird für Mehrgrößensysteme  verwendet. Im Eingrößenfall wird der Durchgriff $d$ skalar und anstelle von Matrizen bleibt ein Eingangsvektor $\jv b$ und ein Ausgangsvektor $\jv c^T$. Mit dem Verzögerungsoperator $z$ kann die entsprechende Transformation der Zustandsgleichung mit
+ein lineare, diskretes ZRM mit Zustandsübergangsmatrix $\jv A_d$ in der Zustandsgleichung und entsprechenden Ein- und Ausgangsmatrizen. Die Auswertung erfordert immer die Kenntnis des Anfangszustands $\jv x_0$. Die hier verwendete Schreibweise umfasst offensichtlich Mehrgrößensysteme mit Ein- und Ausgangsvektoren $\jv u, \jv y$. Im Eingrößenfall wird der Durchgriff $d$ skalar und anstelle von Matrizen bleibt ein Eingangsvektor $\jv b$ und ein Ausgangsvektor $\jv c^T$. Umgekehrt kann man die Beschreibung mit Übertragungs- oder Gewichtsfunktionen leicht auf Matrizen für den Mehrgrößenfall erweitern, deren Elemente jeweils Eingrößenbeschreibungen darstellen (z. B. $G_{2,3}(s)=Y_3(s)/U_2(s)$).
+
+Die Verschiebungseigenschaft der Z-Transformation zeigt sich bei Anwendung z. B. auf $x(k+1)$,
+\begin{align*}
+\jv x(k+1) \laplace zX(z)~~,
+\end{align*}
+identifiziert $z^{-1}$
+als Verzögerungsoperator, welcher dementsprechend im Signalflussgraphen digitaler Filter als Speichersymbol für eine Zeitschritt Verwendung findet.
+
+\begin{tikzpicture}[node distance=2.5cm,auto,>=latex', scale=3]
+    \node [int] (a) {$z^{-1}$};   %\node [int, pin={[init]above:$v_0$}] (a) {$\int$};
+    \node (b) [left of=a,node distance=2cm, coordinate] {a};
+    \node (c) [right of=a] {$$}; %\node [int, pin={[init]above:$p_0$}] (c) [right of=a] {$\int$};
+    \node (d) [right of=c] {$$};
+    \node (e) [right of=d] {$$};
+    \node [coordinate] (end) [right of=c, node distance=2cm]{};
+    \path[->] (b) edge node {$u(k)$} (a);
+    \path[->] (a) edge node {$u(k-1)$} (c);
+\end{tikzpicture} 
+
+Damit kann die entsprechende Transformation der Zustandsgleichung des diskreten ZRM durchgeführt werden,
 \begin{align*}
-x(x+1) \laplace zX(z)
+ z\cdot  \jv X(z)=\jv A  \jv X(z)  \jv B\jv U(z) ~~,
 \end{align*}
-analog zum kontinuierlichen ZRM durchgeführt werden, um die Übertragungsfunktion und die statische Verstärkung 
+
+nach $\jv X(z)$ aufgelöst und in die Ausgabegleichung eingesetzt werden, um die Übertragungsfunktion und die statische Verstärkung 
 \begin{align*}
- {\jv G}(z) &= \jv C(z\jv I - A_k)^{-1} \jv B_k +\jv D;~~\jv K_s=\jv G(1) = -\jv C \jv (\jv I - A)^{-1} \jv B +D 
+ {\jv G}(z) &= \jv C(z\jv I - \jv A)^{-1} \jv B +\jv D;~~~~\jv K_s=\jv G(1) = -\jv C \jv (\jv I - \jv A)^{-1} \jv B +\jv D 
 \end{align*}
-zu ermitteln. Diese ergibt sich im diskreten Fall für $z=e^{0\cdot T}=1$. Übertragungs- oder Gewichtsfunktionen können im Mehrgrößenfall einfach als Matrizen geschrieben werden.
-%\subsection{Wissenschaftliches Schreiben%
- %           \label{sec:Grundlagen-Schreiben}}
+zu ermitteln. Letztere unterscheidet sich formal da im diskreten Fall der statische Zustand für $z=e^{0\cdot T}=1$ oder aus $\jv x(k+1) = \jv x(k)$ ermittelt wird. 
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Halteglieder (ZOH und FOH) in Theorie und Praxis%
+         \label{sec:GrundlagenHalteglieder}}
+					
+\textbf{ZOH (Zero-Order-Hold).} Um mit einer praktischen, algorithmischen Beschreibung zu beginnen, sei das Verhalten des ZOH-Block aus MATLAB/Simulink\texttrademark beschrieben: Der ZOH-Block nimmt zu Beginn der Abtastperiode den Eingangswert und hält ihn konstant für eine Periodendauer $T$. Bezogen auf ein kontinuierliches Eingangssignal $u(t)$ wird damit ein Abtast-Halteglied (sample-and-hold) realisiert. Ein kontinuierliches Eingangssignal $u(t)$ wird in ein stufenförmiges, aber kontinuierliches Signal $y(t)$ in transformiert. Die Höhe der Stufen ist durch die Werte von $u(t)$ zu den Abtastzeitpunkten $kT$ gegeben und wird für eine Periode $T$ gehalten. Das stufenförmige Signal $y(t)$ stimmt \textit{links} mit $u(t)$ überein. Die Hintereinander-Schaltung zweier (oder mehrerer) ZOH-Blöcke mit identischer Abtastperiode ist dementsprechend völlig wirkungslos. Zum Abtastzeitpunkt propagiert der Eingangssignalwert durch alle Blöcke und wird am Ausgang für eine Periode gehalten.	
+
+\begin{tikzpicture}[node distance=2.5cm,auto,>=latex', scale=3]
+    \node [int] (a) {$z^{-1}$};   %\node [int, pin={[init]above:$v_0$}] (a) {$\int$};
+    \node (b) [left of=a,node distance=2cm, coordinate] {a};
+    \node (c) [right of=a] {$$}; %\node [int, pin={[init]above:$p_0$}] (c) [right of=a] {$\int$};
+    \node (d) [right of=c] {$$};
+    \node (e) [right of=d] {$$};
+    \node [coordinate] (end) [right of=c, node distance=2cm]{};
+    \path[->] (b) edge node {$u(k)$} (a);
+    \path[->] (a) edge node {$u(k-1)$} (c);
+\end{tikzpicture} 
+				
+Der Abtastvorgang wurde bereits mathematisch als Umwandlung in eine gewichtete Impulsfolge beschrieben. Das Halteglied nullter Ordnung (Zero-Order-Hold) kann eigentlich nur sinnvoll mit einer solchen Impulsfolge gespeist werden, um den Wichtungsfaktor am Ausgang über eine Periode zu halten. Das Halteglied ohne den Abtaster überträgt also ein (quasi-) kontinuierliches Signal, sinnvollerweise eine Impulsfolge, in ein kontinuierliches, stufenförmiges Signal. Die Impulsantwort 
+ist damit
+\begin{align*}
+g_{ZOH}(t) = \sigma(t) - \sigma(t-T) \laplace G_{ZOH}(s) = \frac{1-e^{-sT}}{s}
+\end{align*}
+
+MATLAB/Simulink\texttrademark übernimmt für den Anwender die Signaltransformation. Ein diskretes System mit kontinuierlichem Eingang wird automatisch mit dem abgetasteten Signal gespeist, wobei zu einer diskreten Folge übergegangen wird. Am Ausgang wird das Signal, wenn es nicht diskret weiterverarbeitet wird implizit entsprechend einem Halteglied nullter Ordnung stufenförmig für die Abtastzeit konstant gehalten.
 
+\textbf{FOH (First-Order-Hold).}  
\ No newline at end of file
diff --git a/3_Euler-Diskretisierung.tex b/3_Euler-Diskretisierung.tex
index 5fb5fc0..039504b 100644
--- a/3_Euler-Diskretisierung.tex
+++ b/3_Euler-Diskretisierung.tex
@@ -1,10 +1,10 @@
-\section{Euler-Diskretisierung
+\subsection{Euler-Diskretisierung
          \label{sec:Euler-Diskretisierung}}
 
 Der Euler-Ansatz besteht darin, die kontinuierliche Ableitung durch den Differenzenquozienten zu ersetzen, im ersten Fall (nicht-kausal) mit dem zukünftigen Wert des abzuleitenden Signals, im zweiten Fall mit dem vergangenen Wert (kausal). In beiden Fällen erhält man kausale, diskrete Systeme, allerdings mit ungünstigen Eigenschaften bezüglich Transformation der Eigenwerte/Pole. Im Video \citep{yawpitchroll2013} sind diese Eigenschaften dargestellt. Von Vorteil ist, dass auch nichtlineare Differenzialgleichungen auf diese Art diskretisiert werden können. Für lineare Systeme könnte der Ansatz als naiv betrachtet, weshalb in der Quelle \citep{yawpitchroll2013} gegen Ende zur Tustin-Approximation Kap. \ref{sec:Tustin-Approximation} übergegangen wird.  
 
    
-\subsection{Euler-Forward-Diskretisierung (explizit)
+\subsubsection{Euler-Forward-Diskretisierung (explizit)
             \label{sec:Euler-Forward-Diskretisierung}}
 
 Der Ansatz der Euler-Diskretisierung ist es, die Differenziale durch den Differenzenquotienten zu ersetzen. Bei der Euler-Forward-Methode
@@ -127,7 +127,7 @@ G_k(s=0) = \jv D- \jv C\jv A_k^{-1} \jv B_k &\stackrel{?}{=}  \jv D+ \jv C(\jv I
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\subsection{Euler-Backward-Diskretisierung (implizit)
+\subsubsection{Euler-Backward-Diskretisierung (implizit)
             \label{sec:Euler-Forward-Diskretisierung}}
 
 Der Ansatz Euler-Backward bedeutet, einen kausalen diskreten Differenziator zu bilden, bei dem die Zeitableitung durch den Differenzenquozienten aus dem aktuellen und dem \textbf{vergangenen} Vorgängerwert des Signals gebildet wird:
diff --git a/4_ZOH-Diskretisierung.tex b/4_ZOH-Diskretisierung.tex
index 594cb15..1e45591 100644
--- a/4_ZOH-Diskretisierung.tex
+++ b/4_ZOH-Diskretisierung.tex
@@ -1,4 +1,4 @@
-\section{ZOH-Diskretisierung
+\subsection{ZOH-Diskretisierung
          \label{sec:ZOH-Diskretisierung}}
 
 \textbf{Sprungantwort.} Im Zeitbereich interpretiert, bedeutet die ZOH-Diskretisierung die Abtastung der Sprungantwort eines Systems. Wie in Bild \ref{fig:Sprungantwort} bereits zusehen ist, liegen die Abtastwerte der diskreten Sprungantwort des ZOH-diskretisierten Systems auf der kontinuierlichen Sprungantwort des Originalsystems. Für alle anderen Ansätze der Diskretisierung gilt das nicht. Das liegt an der Forderung:
@@ -22,10 +22,6 @@ Dort wurde jede Integration, z. B. in einer Blockschaltbild-Darstellung der Diff
 Die Integration wird in Bild x.y veranschaulicht. Die Verzögerung legt nahe, dass es bessere Ansätze gibt, ein kontinuierliches dynamische System zu approximieren, konkret eine Integration mit Trapez-Regel (siehe Abschn. \ref{sec:Tustin-Approximation}).
 
 
-\textbf{ZOH.} Warum heißt dieser  Ansatz Zero-Order-Hold (ZOH)-Approximation? Betrachten wir deshalb zunächst den ZOH-Operator. Dieser exisitiert zunächst in einem kontinuierlichen Sinn. Ein kontinuierliches Eingangssignal $u(t)$ wird in ein stufenförmiges, aber kontinuierliches Signal $y(t)$ in transformiert. Die Höhe der Stufen ist durch die Werte von $u(t)$ zu den Abtastzeitpunkten $kT$ gegeben und wird für eine Periode $T$ gehalten. Das stufenförmige Signal $y(t)$ stimmt \textit{links} mit $u(t)$ überein. Man kann das Verhalten als Sample-and-Hold verstehen.
-
-In MATLAB/Simulink\texttrademark nimmt der ZOH-Block zu Beginn der Periode den Eingangswert und hält ihn konstant für eine Periode $T$. Die Hintereinander-Schaltung zweier (oder mehrerer) ZOH-Blöcke mit identischer Abtastperiode ist dementsprechend völlig wirkungslos. Zum Abtastzeitpunkt propagiert der Eingangssignalwert durch alle Blöcke und wird am Ausgang für eine Periode gehalten.
-
 \textbf{Zeitbereichsäquivalenz.} Regt man ein kontinuierliches System $G_k$ mit einem stufenförmigen, durch ein Abtast-Halteglied (ZOH-Block), gefiltertes Signal an, dann entspricht dieser Eingang einer Aneinanderreihung von gewichteten Sprungfunktionen.
 \[
 u^*(t) = \sum_{k=0}^{\infty} \sigma(t-kT)\cdot (\rho(k)-\rho(k-1))
@@ -62,7 +58,12 @@ Darin stellt $\e^{\jv A}$ eine Matrixexponetialfunktion dar:
 \begin{align*}
 e^{\jv A_k T} = \sum_{i=0}^{\infty} \frac{\jv A^i T^i}{i!}
 \end{align*}
-Ihre numerische Berechnung ist die Hauptaufgabe bei der Ermittlung des diskreten Modells.
+Ihre numerische Berechnung ist die Hauptaufgabe bei der Ermittlung des diskreten Modells. Eine inverse Transformation
+\begin{align*}
+ \jv A_d =e^{\jv A_k T}  ~~\Leftrightarrow~~  \jv A_c = \frac{1}{T} \ln \jv A_d
+\end{align*}
+ist jederzeit möglich, wobei auch der Matrixlogarithmus nicht im Sinne einer elementweisen Operation, sondern als Matrixoperation z. B. in MATLAB\texttrademark implementiert ist.
+
 
 %\begin{minipage}[b]{7cm}
 %  \centering
diff --git a/5_Impulse-Diskretisierung.tex b/5_Impulse-Diskretisierung.tex
index 713e982..7431ef1 100644
--- a/5_Impulse-Diskretisierung.tex
+++ b/5_Impulse-Diskretisierung.tex
@@ -1,4 +1,4 @@
-\section{Impulse-Diskretisierung
+\subsection{Impulse-Diskretisierung
          \label{sec:Impulse-Diskretisierung}}
 
 \textbf{Impulsantwort.} Mit Hilfe der Impulsantwort wird ein lineares System eindeutig charakterisiert. Nimmt man die Impulsantwort eines Systems zur Ermittlung eines diskreten Modells mit der Abtastzeit $T$, dann lautet der Ansatz
diff --git a/6_FOH-Diskretisierung.tex b/6_FOH-Diskretisierung.tex
index fcb1f19..40885c5 100644
--- a/6_FOH-Diskretisierung.tex
+++ b/6_FOH-Diskretisierung.tex
@@ -1,9 +1,48 @@
-\section{FOH-Diskretisierung
+\subsection{FOH-Diskretisierung
          \label{sec:FOH-Diskretisierung}}
 
 
 
 
+
+\textbf{Zutandsraummodell.} Betrachtet man die Rechenvorschrift zur Ermittlung eines diskreten ZRM aus dem kontinuierlichen ZRM erkennt man, dass in der Regel eine Durchgriff $\jv D_d \neq \jv 0$ entsteht, obwohl das kontinuierliche System keinen Durchgriff $\jv D_k = \jv 0$ hat. Es ergibt sich außerdem die gleiche Systemmatrix und damit die gleichen Eigenwerte, wie die folgende Zusammenfassung zeigt:
+
+\begin{empheq}[box=\fbox]{align*}
+%\boxed{
+ \jv A_d &=e^{\jv A_k T}         ~~&~~ \jv B_d &=(\jv A_d(\jv A_d - \jv A_k T - I)(\jv A_k T)^{-2} \jv B_k T +(\jv A_k T \jv A_d - \jv A_d + \jv I)(\jv A_k T)^{-2} \jv B_k T)\\
+%&&= (e{\jv A_k T}-\jv I){\jv A_k}^{-1}\jv B_k\\^
+  \jv C_d &=\jv C_k              ~~&~~ \jv D_d &= \jv (\jv C_d (\jv A_d - \jv A_k T - I)(\jv A_k T)^{-2} \jv B_k T \jv D_k)
+%}
+\end{empheq} 
+
+In \cite{BINGULAC1992293} sind die hier wiedergegebene Ergebnisse für reguläre Matrizen $\jv A_k$ zu finden, wobei der Artikel sich des Weiteren mit dem Spezialfall nicht-regulärer Matrizen beschäftigt. Für $\jv A$ regulär gilt:
+
+\begin{align}\label{ZRM_komisch}
+\jv x_{k+1}&=\jv A_d \jv x_k+\jv G_0 \jv u_k + \jv G_1 \jv u_{k+1},~~\jv x_0\\
+\jv y_k &= \jv C_d \jv x_k + \jv D_d \jv u_k\nonumber
+\end{align} 
+mit
+\begin{align*}
+\jv G_0 &= \jv M_0 \jv B_k T\\
+\jv M_0 &= (\jv A_k T \jv A_d - \jv A_d + \jv I)(\jv A_k T)^{-2}\\
+\jv G_1 &= \jv M_1 \jv B_k T\\
+\jv M_1 &= (\jv A_d - \jv A_k T - I)(\jv A_k T)^{-2}\\
+\end{align*}
+Das System \ref{ZRM_komisch} kann genau wie  .. durch Aufspalten der Zustandsgleichung
+\begin{align*}
+\jv x = \jv x^1+ \jv x^2:& \jv x^1_{k+1}&= \jv A_d \jv x_k + \jv G_0 \jv u_{k+1}\\
+             & \jv x^2_{k+1}&= \jv G_1 \jv u_{k+1}
+\end{align*}
+in ein System mit dem neuen Zustandsvektor $\tilde{\jv x} = \jv x^1$ umgeformt werden
+\begin{align*}
+\tilde{\jv x}_{k+1}&=\jv A_d \tilde{\jv x}_k+ \overbrace{(\jv A_d\jv G_1 +\jv G_0)}_{\jv B_d} \jv u_{k}\\
+\jv y_k &= \jv C_d \tilde{\jv x}_k + \underbrace{(\jv C_d \jv G_1 \jv D_k)}_{\jv D_d} \jv u_k\nonumber
+\end{align*} 
+
+Hiermit können die angegebenen Matrizen gewonnen werden (wie im MATLAB\texttrademark-Skript verifiziert).
+
+
+
 %\begin{eqnarray}
 %  P_\text{Welle}         &=& 2  \pi  M  n \label{eq:drehmoment-1} \\
 %  \Leftrightarrow\quad M &=& \frac{P_\text{Welle}}{ 2  \pi  n} \label{eq:drehmoment-2}
diff --git a/7_Tustin-Diskretisierung.tex b/7_Tustin-Diskretisierung.tex
index f935766..4d0f098 100644
--- a/7_Tustin-Diskretisierung.tex
+++ b/7_Tustin-Diskretisierung.tex
@@ -1,4 +1,4 @@
-\section{Tustin-Approximation / Trapez-Regel
+\subsection{Tustin-Approximation / Trapez-Regel
          \label{sec:Tustin-Approximation}}
 
 \textbf{Integrationsmethode.} Die Tustin-Approximation kann mit der Integration nach der Trapez-Regel identifiziert werden. Betrachtet man die bisherigen Integrationsansätze, drängt sich ein verbesserter Ansatz für die Flächenberechnung zwischen zwei Zeitpunkten mit Hilfe der Trapez-Regel auf. Darüber wird aus dem Wert $I(k)$ rekursiv der nächste Wert wie folgt berechnet:
@@ -27,11 +27,24 @@ als Übertragungsfunktion für den Akkumulator.
 \end{tikzpicture}
 
 Es gilt die Entsprechung 
-\begin{align*}
+\begin{align}\label{eq:sTustin}
 s= \frac{2}{T}   \frac{z-1}{z+1}~~~\text{bzw.}~~~z= \frac{s+2/T}{2/T-s} 
-\end{align*}
+\end{align}
 folgt
 
+\textbf{Reihenentwicklung}. Aus einer anderen Perspektive hängen s-Ebene und z-Ebene nach \eqref{eq:sz} wie folgt zusammen
+\begin{align*}
+z = e^{sT} \Leftrightarrow s=\frac{1}{T} \ln(z)~~.
+\end{align*}
+Setzt man jedoch den Ausdruck für $s$ so in eine Übertragungsfunktion ein, erhält man keinen Term, der eine lineare z-Übertragungsfunktion darstellt. Laut \cite{unbehauen2000RT2} kann der natürliche Logarithmus in eine Reihe entwickelt werden
+\begin{align*}
+s &= \frac{1}{T}2\left(\frac{z-1}{z+1} + \frac{1}{3} (\frac{z-1}{z+1})^3 + \frac{1}{5}(\frac{z-1}{z+1})^5\hdots \right) \\
+&\approx \frac{2}{T}\frac{z-1}{z+1}
+\end{align*}
+Die Darstellung lässt sich aus der bekannten Taylor-Reihe des natürlichen Logarithmus herleiten, was in Anhang \ref{anh:Logarithmus} verschoben wurde.
+Durch Abbruch nach dem ersten Summanden entsteht der bekannte Zusammenhang \eqref{eq:sTustin}.
+
+
 \textbf{Differenziation.} Die beiden Euler-Methoden aus Kap. \ref{sec:Euler-Diskretisierung} wurden dardurch motiviert, dass der Differzenquotient die Ableitung genau an den Abtastzeitpunkten approximiert. Versuchen wir stattdessen die Approximation genau zwischen zwei Abtastzeitpunkten also für die Ableitung
 \begin{align*}
 \dot x_k((k-\frac{1}{2}) T) \approx \frac{x(k)-x(k-1)}{T} 
@@ -46,10 +59,39 @@ approximiert wird, dann gitl für den den Differenziator $y=\dot u$ die Diskreti
 \end{align*}
 Der kausale Differenzierer ist also durch
 \begin{align*}
-y(k=)\frac{2}{T}(x(k)-x(k-1))-y(k-1);~~G_D(z) = \frac{2}{T} \frac{z-1}{z+1}
+y(k)=\frac{2}{T}(x(k)-x(k-1))-y(k-1);~~G_D(z) = \frac{2}{T} \frac{z-1}{z+1}
 \end{align*}
 gegeben.
 
+\textbf{Zustandsraummodell.} In \cite{WolframVogt} wird das ZRM im z-Bereich aus der kontinuierlichen Zustandsgleichung im Frequenzbereich
+\begin{align*}
+ s{\jv X}(s) &= \jv A_k \jv X(s) + \jv B_k \jv U(s) \overbrace{\rightarrow}_{\text{Tustin}} \frac{2}{T} \frac{z-1}{z+1}{\jv X}(z) &= \jv A_k \jv X(z) + \jv B_k \jv U(z)
+\end{align*}
+über den Ansatz \eqref{eq:sTustin} hergeleitet. Da ZRM-Darstellungen nicht eindeutig sind, werden dort mehrere Formen für den Spezialfall $\jv D= \jv 0, \jv C=\jv I$ berechnet. Im allgemeinen Fall gilt
+\begin{align*}
+ \frac{z-1}{z+1}{\jv X}(z) &= \underbrace{\frac{T}{2}\jv A_k}_{\tilde{\jv A}} \jv X(z) + \underbrace{\frac{T}{2}\jv B_k}_{\tilde{\jv B}} \jv U(z)\\
+(\jv I-\tilde{\jv A})z \jv X(z) &= (\jv I+\tilde{\jv A})\jv X(z) + \tilde{\jv B} (z+1) \jv U\\
+z \jv X(z) &= \underbrace{(\jv I-\tilde{\jv A})^{-1}(\jv I+\tilde{\jv A})}_{\jv A_d}\jv X(z) + (\jv I-\tilde{\jv A})^{-1}\tilde{\jv B} (z+1) \jv U\\
+\Laplace\\
+\jv x_{k+1} &= \jv A_d \jv x_k + (\jv I-\tilde{\jv A})^{-1}{\tilde{\jv B}} (\jv u_{k+1} + \jv u_k)
+\end{align*}
+
+Um auf ZRM-Form zu kommen, folgen wir dem Ansatz x.y, welcher hier einer Zustandstransformation
+y.z
+entspricht und erhalten
+%\boxed{
+\begin{empheq}[box=\fbox]{align*}
+%\boxed{
+ \jv A_d &=(\jv I-\tilde{\jv A})^{-1}(\jv I+\tilde{\jv A})        ~~&~~ \jv B_d &= ((\jv I-\tilde{\jv A})^{-1}(\jv I+\tilde{\jv A}) +\jv I)(\jv I-\tilde{\jv A})^{-1} \tilde{\jv B} \\
+%&&= (e{\jv A_k T}-\jv I){\jv A_k}^{-1}\jv B_k\\^
+  \jv C_d &=\jv C_k              ~~&~~ \jv D_d &= \jv C_k(I-A)^{-1} \jv B
+%}
+\end{empheq}
+%}
+
+Diese Darstellung wird scheinbar nicht im MATLAB\texttrademark-Algorithmus verwendet, da sich jenseits der Systemmatrix unterschiedliche Transformationsergebnisse ergeben. Im Skript Link x.y wird aber gezeigt, dass die Systeme identisch sind.
+
+
 
 %\begin{eqnarray}
 %  P_\text{Welle}         &=& 2  \pi  M  n \label{eq:drehmoment-1} \\
diff --git a/8_Matched-Filter.tex b/8_Matched-Filter.tex
index f36c2fa..20ca497 100644
--- a/8_Matched-Filter.tex
+++ b/8_Matched-Filter.tex
@@ -1,9 +1,9 @@
-\section{Matched-Filter / Pol-/Nullstellen-Approximation 
+\subsection{Matched-Filter / Pol-/Nullstellen-Approximation 
          \label{sec:Matched-Filter}}
 
 Ein Ansatz der Approximation ist es, Pole und Nullstellen entsprechend der Transformation
 \begin{align}
-z_{P_i} = e^{z_{P_i} T}
+z_i = e^{s_i T}
 \end{align}
 für das diskrete System festzulegen. 
 
diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf
index 50df5aac65ec5c861ffbcbfe1eb597509fbf6fcb..22b186360271d0c056919dd745a7df205076c02a 100644
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diff --git a/Matlab/Diskretisierung_FOH.m b/Matlab/Diskretisierung_FOH.m
new file mode 100644
index 0000000..6c987e1
--- /dev/null
+++ b/Matlab/Diskretisierung_FOH.m
@@ -0,0 +1,48 @@
+% Diskretisierung linearer Systeme nach FOH
+clear all; close all
+
+%% System zweiter Ordnung ohne Durchgriff, stabil, komplexe Eigenwerte -> überschwingend
+sys = tf([1,2,3],[1 1 1]);
+figure(3);pzmap(sys)
+[p_sys, z_sys]=pzmap(sys);
+
+% State space Modell, State Space ohne Durhcgriff
+stsp_sys = ss(sys)
+%stsp_sys2 = ss(stsp_sys.A,stsp_sys.B,stsp_sys.C,0)
+
+%% Diskretisierung des ZRM
+T=0.1;
+sys_FOHmatlab = c2d(stsp_sys,T,'FOH') 
+
+% eigene Berechnung via Matrizen 
+A_c =stsp_sys.A; B_c =stsp_sys.B; C_d =stsp_sys.C; D_c =stsp_sys.D; % aus Modell
+I=eye(size(A_c,1))  % Einheitsmatrix in reichtiger Dimension
+
+% Systemmatrix (exakt wie bei ZOH)
+A_d = expm(A_c * T);
+% A_c= 1/T * logm(A_d)  % inverse Transformation 
+
+M0 = (A_c*T*A_d-A_d + I) * (A_c*T)^(-2)
+G0 = M0*B_c* T
+
+M1 = (A_d-A_c * T - I) * (A_c*T)^(-2)
+G1 = M1*B_c* T
+
+B_d =A_d*G1+G0
+D_d = C_d * G1 +D_c
+
+sys_FOHmatrix = ss(A_d, B_d, C_d, D_d, T)
+
+%% Vergleich
+if (max(max(sys_FOHmatlab.A - A_d)) > eps) | (max(max(sys_FOHmatlab.B - B_d)) > eps) | (max(max(sys_FOHmatlab.C - C_d)) > eps) | (max(max(sys_FOHmatlab.D - D_d)) > eps)
+    disp('diskretisierung ungleich');
+    pause;
+end
+
+%% Pol/Nullstellenvergleich
+figure(4);pzmap(sys_FOHmatlab);
+%[y,t]=impulse(sys_EF); plot(t, y ,'*','color', 'magenta')
+%legend({'','ZOH','FOH','Impulse','Tustin'},'Location','North')
+
+% Bild speichern
+%saveas(gcf,'../bilder/EF_Vergleich.png')
diff --git a/Matlab/Diskretisierung_Tustin.m b/Matlab/Diskretisierung_Tustin.m
new file mode 100644
index 0000000..5e3af53
--- /dev/null
+++ b/Matlab/Diskretisierung_Tustin.m
@@ -0,0 +1,44 @@
+% Diskretisierung linearer Systeme nach Tustin
+clear all; close all
+
+%% System zweiter Ordnung ohne Durchgriff, stabil, komplexe Eigenwerte -> überschwingend
+sys = tf([1,1],[1 1 1]);
+figure(3);pzmap(sys)
+[p_sys, z_sys]=pzmap(sys);
+
+% State space Modell, State Space ohne Durhcgriff
+stsp_sys = ss(sys)
+%stsp_sys2 = ss(stsp_sys.A,stsp_sys.B,stsp_sys.C,0)
+
+%% Diskretisierung des ZRM
+T=0.1;
+sys_TTmatlab = c2d(stsp_sys,T,'Tustin') 
+
+% eigene Berechnung via Matrizen 
+A_c =stsp_sys.A; B_c =stsp_sys.B; C_d =stsp_sys.C; D_c =stsp_sys.D; % aus Modell
+I=eye(size(A_c,1))  % Einheitsmatrix in reichtiger Dimension
+A_cT=T/2 *A_c; B_cT=T/2 *B_c;
+
+% Systemmatrix (exakt wie bei ZOH)
+A_d = inv(I-A_cT) * (I+A_cT) ;
+B_d = (A_d + I) * inv(I-A_cT) * B_cT
+D_d = C_d *inv(I-A_cT) * B_cT +D_c
+sys_TTmatrix = ss(A_d, B_d, C_d, D_d, T)
+
+%% Vergleich, Matlab-Algorithmus und Matrizen
+%% Systeme sind offensichtlich verschieden, aber A_d gleich
+% if (max(max(sys_FOHmatlab.A - A_d)) > eps) | (max(max(sys_FOHmatlab.B - B_d)) > eps) | (max(max(sys_FOHmatlab.C - C_d)) > eps) | (max(max(sys_FOHmatlab.D - D_d)) > eps)
+%     disp('diskretisierung ungleich');
+%     pause;
+% end
+
+%% Pol/Nullstellenvergleich, Systemantwort
+figure(4);pzmap(sys_TTmatlab);hold on; pzmap(sys_TTmatrix);
+%
+figure(1)
+[y,t]=impulse(sys_TTmatlab); plot(t, y ,'*','color', 'magenta')
+hold on
+[y,t]=impulse(sys_TTmatrix); plot(t, y ,'*','color', 'blue')
+
+% Bild speichern
+%saveas(gcf,'../bilder/?.png')
diff --git a/anhang.tex b/anhang.tex
index b3a64e4..c3fa122 100644
--- a/anhang.tex
+++ b/anhang.tex
@@ -52,9 +52,39 @@ Benennt man jetzt den Zustand $\jv x_1 = \tilde{\jv x}$ um, kann man die Matrize
 
 
 % Anhang C
-\section{Datenblätter und Zeichnungen}
-\label{anh:datenblaetter}
+\section{Spezielle Reihenentwicklung der Logarithmus-Funktion}
+\label{anh:Logarithmus}
+Die Taylor-Reihenentwicklung
+\begin{align*}%\label{ZRM_disk}
+f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}x_0}{n!} (x-x_0)^n
+\end{align*} 
+um den Entwicklungspunkt $x_0$ wird als bekannt vorausgesetzt. Angewendet auf den natürlichen Logarithmus um $x_0=1$
+\begin{align*}%\label{ZRM_disk}
+\ln(x) = 0 + (x-1) -\frac{1}{2} (x-1)^2+\hdots = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} (x-1)^n
+\end{align*} 
+stellt sie den Ausgangspunkt für z. B. in \cite{wiki1} hergeleitete, variierende Darstellungen von Entwicklungen. Dazu bilden wir
+\begin{align*}%\label{ZRM_disk}
+\ln(1+y) &= \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k} y^k
+\ln(1-y) &= \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k} (-y)^k = \sum_{k=1}^{\infty} \frac{(-1)}{k} y^k
+\end{align*}
+Der Variablenwechsel soll dazu dienen, die Struktur in der folgenden Umformung wiederzufinden: 
+\begin{align*}%\label{ZRM_disk}
+\ln(x) = \ln(\frac{2x}{2}) = \ln( \frac{1+\frac{x-1}{x+1}   }{1- \frac{x-1}{x+1}  }) &=  \ln(1+\frac{x-1}{x+1}) - \ln(1-\frac{x-1}{x+1})\\
+&= \ln(1+y) - \ln(1-y)
+\end{align*}
+Vor dem Einsetzen des Bruchs für $y$ bieten sich weitere Umformungen an:
+\begin{align*}%\label{ZRM_disk}
+\ln(x) = \ln(1+y) - \ln(1-y) &= \sum_{k=1}^{\infty} ((-1)^{k-1}+1) \frac{y^k}{k}\\
+&= 2\sum_{m=1}^{\infty}  \frac{y^{2m-1}}{2m-1}\\
+&= 2y\sum_{m=0}^{\infty}  \frac{y^{2m}}{2m+1}\\
+\end{align*}
+und damit wie bereits bekannt
+\begin{align*}%\label{ZRM_disk}
+\ln(x) &= 2\sum_{k=1}^{\infty} \frac{1}{2k-1} \frac{x-1}{x+1}^{2k-1}\\
+&= 2\left( \frac{x-1}{x+1} +\frac{1}{3} \left(\frac{x-1}{x+1}\right)^3 + \frac{1}{5}\left(\frac{x-1}{x+1}\right)^5 +\hdots\right)
+\end{align*}
 
+ 
 
 %%%%%%%%%%%%%%%%
 %\section{Weitere \LaTeX{}-Beispiele, Tikz und Einbinden von csv-Dateien und xls}
diff --git a/deckblatt-cvh.tex b/deckblatt-cvh.tex
index 34f85da..c002289 100644
--- a/deckblatt-cvh.tex
+++ b/deckblatt-cvh.tex
@@ -22,7 +22,9 @@
   von\par
   \bigskip\bigskip
   {\Large\textbf{\theauthor}\par}
-  \smallskip
+  %\smallskip
+	\bigskip\bigskip\bigskip
+  \dokumentname{}\\
  
 
 \end{center}
diff --git a/hauptdatei.bib b/hauptdatei.bib
index ca25472..9ba669d 100644
--- a/hauptdatei.bib
+++ b/hauptdatei.bib
@@ -70,3 +70,20 @@ author = {S. Bingulac and D.L. Cooper},
 abstract = {A general procedure for calculating an equivalent ramp invariant (FOH) continuous-time model corresponding to a given discrete-time model is derived. Previously, only SISO and ΜΙΜΟ systems having distinct non-zero continuous eigenvalues have been treated. In this respect, the presented procedure represents a generalization of published results. An illustrative computational example is included.}
 }
 
+@article{WolframVogt,
+title = {Zeitdiskrete Filteralgorithmen zur Erzeugung zeitlicher Ableitungen},
+journal = {At - Automatisierungstechnik},
+volume = {50},
+number = {7},
+pages = {346},
+year = {2002},
+author = {A. Wolfram und M. Vogt},
+}
+
+@MISC{wiki1,
+author = {Wikipedia},
+title = {Natural logarithm},
+month = {März},
+year = {2021},
+note = {\url{https://en.wikipedia.org/wiki/Natural_logarithm}}
+}
\ No newline at end of file
diff --git a/hauptdatei.tex b/hauptdatei.tex
index 2971cda..1813c59 100644
--- a/hauptdatei.tex
+++ b/hauptdatei.tex
@@ -184,11 +184,10 @@
 %
   %
   \newcommand{\institutname}{  Institut für Elektrotechnik\\  und Informationstechnik}%  \newcommand{\institutname}{  Institut für Informatik\\  und eingebettete Systeme}
-  \newcommand{\thetitle}{Diskretisierung linearer Systeme}
-  \newcommand{\theauthor}{Jan Falkenhain}
-  
+  \newcommand{\thetitle}{Diskretisierung linearer Systeme und Anwendungen in der digitalen Regelung}
+  \newcommand{\theauthor}{Jan Falkenhain}  
   \newcommand{\discipline}{Mechatronik \& Informationstechnologie}
-
+	\newcommand{\dokumentname}{Mit Beispielen in MATLAB/Simulink\texttrademark}
   \newcommand{\finaldate}{\today}
 	\newcommand{\jv}{\pmb}
 
@@ -236,8 +235,13 @@
   \clearpage
   \input{2_Grundlagen}
 
+  % Diskretisierungsmethoden
+	\clearpage
+  \section{Diskretisierungsmethoden
+         \label{sec:Diskretisierungsmethoden}}
+
   % Abschnitt 3: Euler
-  \clearpage
+  %\clearpage
   \input{3_Euler-Diskretisierung}
 	
 	% Abschnitt 7: Tustin
@@ -259,6 +263,12 @@
 % Abschnitt 7: Beispiele
   \clearpage
   \input{8_Matched-Filter}
+	
+	% Diskretisierungsmethoden
+	\clearpage
+  \section{Anwendungen in der digitalen Regelung
+         \label{sec:Regelung}}
+	\input{Regelungstechnik}
 
  % Abschnitt 4: Zusammenfassung
 %  \clearpage
-- 
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