From f03b371265f150b32cc6591600ce6ec5f33b4ecf Mon Sep 17 00:00:00 2001
From: Jan Falkenhain <jan.falkenhain@hs-bochum.de>
Date: Fri, 23 Apr 2021 18:41:17 +0200
Subject: [PATCH] 10erd commit

---
 1_Einleitung.tex                              |   4 +-
 2_Grundlagen.tex                              |   4 +-
 4_ZOH-Diskretisierung.tex                     | 118 +++++++++++++----
 5_Impulse-Diskretisierung.tex                 |  32 ++++-
 6_FOH-Diskretisierung.tex                     | 121 +++++++++++-------
 FB_Diskretisierung.pdf                        | Bin 1100969 -> 1471218 bytes
 Regelungstechnik.tex                          |  72 +++++++++++
 anhang.tex                                    |  18 ++-
 bilder/PIDmod.PNG                             | Bin 0 -> 169903 bytes
 "bilder/zeitbereichs\303\244quivalenzFOH.PNG" | Bin 0 -> 130186 bytes
 "bilder/zeitbereichs\303\244quivalenzZOH.PNG" | Bin 0 -> 112253 bytes
 hauptdatei.bib                                |  23 +++-
 12 files changed, 309 insertions(+), 83 deletions(-)
 create mode 100644 bilder/PIDmod.PNG
 create mode 100644 "bilder/zeitbereichs\303\244quivalenzFOH.PNG"
 create mode 100644 "bilder/zeitbereichs\303\244quivalenzZOH.PNG"

diff --git a/1_Einleitung.tex b/1_Einleitung.tex
index 617df05..8d72153 100644
--- a/1_Einleitung.tex
+++ b/1_Einleitung.tex
@@ -30,7 +30,7 @@ Besonders einfach sind die Ansätze der Euler-Diskretisierung \ref{sec:Euler-Dis
       \hline
       Forward-Euler             & Ja 		&  $s\equiv (z-1)/T$     		 &$\displaystyle T\frac{1}{z-1}$ &    -          &$z_i =s_iT + 1$ 			&$z_{i0} =s_{i0}T + 1$& $=G_k(0)$\\
       \hline
-	Backward-Euler           & Ja  		&  $s\equiv \frac{z-1}{zT}$      &$\displaystyle \frac{Tz}{z-1}$ & -             &$z_i = \frac{1}{1-s_iT}$   & \begin{tabular}{@{}c@{}}$z_{i0}\ = \frac{1}{1-s_{i0}T} ~\&~ (m-n)$\\Nullstellen bei $z_{i0}=0$\end{tabular}
+	Backward-Euler           & Ja  		&  $s\equiv \frac{z-1}{zT}$      &$\displaystyle T\frac{z}{z-1}$ & -             &$z_i = \frac{1}{1-s_iT}$   & \begin{tabular}{@{}c@{}}$z_{i0}\ = \frac{1}{1-s_{i0}T} ~\&~ (m-n)$\\Nullstellen bei $z_{i0}=0$\end{tabular}
 			%$z_i = \frac{1}{1-s_iT} +(m-n)$ Nullstellen bei $z_{i0}=0$  
 			& $=G_k(0)$ \\
 			\hline
@@ -40,7 +40,7 @@ Besonders einfach sind die Ansätze der Euler-Diskretisierung \ref{sec:Euler-Dis
       \hline
 	Impuls         & Nein    &	- 		&$\displaystyle \frac{z}{z-1}$	 			&g(kT)=g(t)&$z_i=e^{s_i T}$ &  &$\displaystyle \lim_{T->0}G_d(1)=G_k(0)$\\
 			\hline
-	ZOH   										& Nein	&-  &		$\displaystyle \frac{Tz}{z-1}$ 		&    h(kT)=h(t)&$z_i=e^{s_i T}$& &$=G_k(0)$\\
+	ZOH   										& Nein	&-  &		$\displaystyle T\frac{1}{z-1}$ 		&    h(kT)=h(t)&$z_i=e^{s_i T}$& &$=G_k(0)$\\
       \hline
 	FOH       		     &	Nein	&-	 	& $\displaystyle \frac{T}{2} \frac{z+1}{z-1}$ 	&Rampe $r(kT)=r(t)$  &$z_i=e^{s_i T}$&  &$=G_k(0)$\\
       \hline
diff --git a/2_Grundlagen.tex b/2_Grundlagen.tex
index 9721b82..f9a4b50 100644
--- a/2_Grundlagen.tex
+++ b/2_Grundlagen.tex
@@ -300,7 +300,9 @@ Wer Übung im Auswerten von Faltungsintegralen hat erkennt, dass die Impulsantwo
 \begin{align*}
 g_{FOH}(t) =\frac{1}{T} g_{ZOH}(t) * g_{ZOH}(t)~~ \laplace~~ G_{ZOH}^2(s) =\frac{1}{T}\cdot G_{FOH}(s) 
 \end{align*}
- mdl. zeigt die Realisierung in MATLAB/Simulink\texttrademark.
+Mögliche Realisierungenin MATLAB/Simulink\texttrademark~zeigt die folgende Datei:
+
+\href{https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/ZOHundFOHblocks.slx}{File on Git: ZOHundFOHblocks.slx}
 
 \textbf{Nicht-kausale Halteglieder.}  In theoretischen Überlegungen spielen nicht-kausale, und damit ja nicht realisierbare, Systeme manchmal eine Rolle.
 Für das symmetrische Halteglied erster Ordnung
diff --git a/4_ZOH-Diskretisierung.tex b/4_ZOH-Diskretisierung.tex
index 931e97d..1c8f62f 100644
--- a/4_ZOH-Diskretisierung.tex
+++ b/4_ZOH-Diskretisierung.tex
@@ -6,9 +6,26 @@
 h_k(kT) =h_d (k)
 \end{align}
 
-\textbf{Integration.} Anhand der Bedingung \ref{ZOH:bed} soll abgeleitet werden, wie der diskrete Integrator, also der Akkumulator im ZOH-Sinn aussieht. Als Reaktion auf die Sprungfunktion liefert der kontinuierliche Integrator eine Rampe mit Steigung eins. Bei $y(0)=0$ liegt ein Knick vor, für $t>0$ beträgt die Steigung von $y(t)=t$ eins (entsprechend der Integratorkonstante). Die Abtastung dieser Sprungantwort liefert
+\textbf{Integration.} Anhand der Bedingung \ref{ZOH:bed} soll abgeleitet werden, wie der diskrete Integrator, also der Akkumulator im ZOH-Sinn aussieht. Als Reaktion auf die Sprungfunktion liefert der kontinuierliche Integrator eine Rampe mit Steigung eins. 
+
+\begin{tikzpicture}%[domain=0:4]
+\draw[-] (-0.1, 1) node[left] {1} -- (0.1,1);
+\draw[color=blue] (0.05,1 )  -- (2,1) node[right] {$u(t)$};
+\draw[-*] (0, -0.1) node[below] {} -- (0,0.1);
+\draw[-*] (0.4, -0.1) node[below] {$T$} -- (0.4,0.5);
+\draw[-*] (0.8, -0.1) node[below] {$$} -- (0.8,0.9);
+\draw[-*] (1.2, -0.1) node[below] {$3T$} -- (1.2,1.3);
+\draw (0, 0)  -- (1.3,1.3);
+
+\draw[->] (-0.2,0) -- (2.2,0) node[right] {$t$};
+ \draw[->] (0,-0.5) -- (0,1.5) node[above right] {$y(t)$};
+\end{tikzpicture}
+
+Bei $y(0)=0$ liegt ein Knick vor, also
+\[y(t)=h(t)=t\sigma(t)~~.\]
+Die Steigung beträgt für $t>0$ eins (entsprechend der Integratorkonstante). Die Abtastung dieser Sprungantwort liefert
 \[
-y(k)= kT = y(k-1) + T~~,\text{für}~k>0
+y(k)= kT = y(k-1) + T~~,\text{für}~k>0, y(0)=0
 \] 
 aber wegen der Sprungfunktion, also $u(k)=1$ für $k\geq 0$
 \[
@@ -55,21 +72,24 @@ Dort wurde jede Integration, z. B. in einer Blockschaltbild-Darstellung der Diff
 \end{figure}
 Die Integration wird in Bild \ref{fig:IntZOH} 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{Diskretisierung der Übertragungsfunktion.} Erinnern wir uns an Korrespondenztabellen mit Laplace- und Z-Transformation, so konnten wir die Übertragungsfunktion $G(s)$ als Transformierte der Impulsantwort $g(t)$ dort finden. Die abgetastete $g(k)=g(kT)$ war dort meist nicht zu finden, dafür direkt deren Z-Transformierte $G(z)$. So konnte die entsprechende Korrespondez der Impuls-Diskretisierung aus der betreffenden Zeile abgelesen werden.\\
+\textbf{Diskretisierung der Übertragungsfunktion.} Erinnern wir uns an Korrespondenztabellen mit Laplace- und Z-Transformation, so konnten wir die Übertragungsfunktion $G(s)$ als Transformierte der Impulsantwort $g(t)$ dort finden. Die abgetastete Folge $g(k)=g(kT)$ war dort meist nicht zu finden, dafür direkt deren Z-Transformierte $G(z)$. So konnte die entsprechende Korrespondenz der Impuls-Diskretisierung aus der betreffenden Zeile abgelesen werden.\\
 
 Für ein System $G(s)$ lautet die Laplace-Transformierte der Sprungantwort
 \begin{align}\label{eq:Hvons}
-H(s) = \frac{1}{s} G(s) \Laplace h(t)
+H(s) = \frac{1}{s} G(s) ~\Laplace~ h(t)
 \end{align}
-Die Abtastfolge $h(k)$ findet man zumeist nicht, wenn man  in der Tabelle nach $H(s)$ entsprechend Gl. \eqref{eq:Hvons} sucht, aber die Z-Transformierte $H(z)$. Über den Zusammenhang zwischen $G(z)$ und $H(z)$ erhalten wir
+Die Abtastfolge $h(k)$ findet man zumeist nicht, wenn man  in der Korrespondenztabelle nach den Termen entsprechend Gl. \eqref{eq:Hvons} sucht, aber die Z-Transformierte $H(z)$. Über den Zusammenhang zwischen $G(z)$ und $H(z)$ erhalten wir
 \[
 H(z) = \frac{z}{z-1}G(z)
 \]
  bzw. in unserem speziellen Fall
 \[
-G_{ZOH}(z) = H(z) \frac{z-1}{z}~~.
+G_{d,ZOH}(z) = H(z) \frac{z-1}{z}~~.
 \]
-
+Zusammenfassend kann die (implizite) Vorgehensweise kompakt ausgedrückt werden:
+\begin{align*}
+G_{d,ZOH}(z)= \frac{z-1}{z}\cdot Z\{ L^{-1}\{ \frac{1}{s} \cdot G(s)\}|_{t=kT} \}~~.
+\end{align*}
 
 \textit{\textbf{Beispiel: Die Übertragungsfunktion des Integrators lautet $G(s)=1/s$. Wir suchen nach der Z-Transformierten der Abtastung von $H(s)=1/s^2$. 
 Tabell \ref{table:impuls} liefert uns
@@ -83,41 +103,55 @@ ermitteln.
 }
 }
 
-\textbf{Zeitbereichsäquivalenz.} Warum heißt dieser Ansatz Zero-Order-Hold (ZOH)-Diskretisierung? 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.
+\textbf{Zeitbereichsäquivalenz.} Warum heißt dieser Ansatz Zero-Order-Hold (ZOH)-Dis\-kretisierung? Regt man ein kontinuierliches System $G_k(s)$ mit einem stufenförmigen, durch ein Abtast-Halteglied (ZOH-Block), gefiltertes Signal an, dann entspricht dieser Eingang einer Aneinanderreihung von gewichteten Sprungfunktionen,
+\[
+u^*(t) = \sum_{k=0}^{\infty} \sigma(t-kT)\cdot (u(k)-u(k-1))~~,
+\]
+wobei die verwendete Differenz aus aufeinanderfolgenden, abgetasteten Eingangswerte der Sprunghöhe dieser Treppenfunktion bei $kT$ entspricht. 
 \[
-u^*(t) = \sum_{k=0}^{\infty} \sigma(t-kT)\cdot (\rho(k)-\rho(k-1))
+u(k)=u(kT) =u*(kT) \forall k\geq 0
 \] 
-Die Gewichtung diese Treppenfunktion wird durch die mit der Periode $T$ abgetasteten Werte 
-$\rho(k)=u(kT) =u*(kT) \forall k\geq 0$ erzeugt. Die Aneinanderreihung (oder Summierung) der Impulse kann als Eingangsfolge
-\[u(k)=\sum_{k=0}^\infty \rho(k) \]
-durch das diskrete System $G_d$ verarbeitet werden. Die Überlagerung von kontinuierlichen Sprungfunktionen $u^*(t)$ führt bei Abbildung mit $G(s)$ auf das Ausgangssignal
+
+\begin{figure}[H]
+\begin{center}
+  \includegraphics[width=0.8\textwidth]{bilder/zeitbereichsäquivalenzZOH}
+      \caption{Reihenschaltung aus ZOH und System. Abgetasteter Eingang, verarbeitet vom ZOH-diskretisierten System liegt auf kontinuierlichen Ausgang. %\citep{yawpitchroll2013}%
+             \label{fig:zeitbereichsäquivalenzZOH}}
+\end{center}
+\end{figure}
+
+Die Eingangsfolge
+$u(k)$ 
+kann durch das diskrete System $G_{d,ZOH}~\laplace~g_d(k)$ verarbeitet werden und die Ausgangsfolge $y(k)$ erzeugen. Die Überlagerung von kontinuierlichen Sprungfunktionen $u^*(t)$ führt bei Abbildung mit $G(s)$ auf das Ausgangssignal
 \begin{align*}
-y^*(t) &= \sum_{k=0}^{\infty} h_k(t-kT)\cdot (\rho(k)-\rho(k-1))
+y^*(t) &= \sum_{k=0}^{\infty} h_k(t-kT)\cdot (u(k)-u(k-1))
 \end{align*}
 mit den Sprungantworten $h_k$ und damit
 \begin{align*}
-y^*(t) &= h_k(t) \rho(0) &+ k_k(t-T) (\rho(1)-\rho(0))\\
-&&+ k_k(t-2T) (\rho(2)-\rho(1))\hdots
-&= \rho(0) (h_k(t)-h_k(t-T))\\
-&+\rho(1) (h_k(t-T)-h_k(t-2T))\\
+y^*(t) &= h_k(t) u(0) + k_k(t-T) (u(1)-u(0))\\
+&~~~~~~~~~~~~~~+ k_k(t-2T) (u(2)-u(1))\hdots\\
+&= u(0) (h_k(t)-h_k(t-T))\\
+&+u(1) (h_k(t-T)-h_k(t-2T))\\
 &\vdots\\
-&= \sum_{k=0}^{\infty} \rho(k)(h_k(t-kT)-h_k(t-[k+1]T))\\
+&= \sum_{j=0}^{\infty} u(j)(h_k(t-jT)-h_k(t-[j+1]T))\\
 \end{align*}
 durch umsortieren. Tasten wird das Signal in dieser Form ab, können wir im Sinne der ZOH-Diskretisierung die diskrete Sprungantwort einsetzen
 \begin{align*}
-y^*(kT) &= \sum_{j=0}^{\infty} \rho(k) \underbrace{(h_k([k-j]T-h_k([k-j-1]T))}_{
+y^*(kT) &= \sum_{j=0}^{\infty} u(k) \underbrace{(h_k([k-j]T)-h_k([k-j-1]T))}_{
 =h_d(k-j)-h_d(k-j-1) = g_d(k-j)
 } \\
-&= \sum_{j=0}^{\infty} \rho(j)g_d(k-j) = \rho(k) \ast g_d(k) = y(k)\\
+&= \sum_{j=0}^{\infty} u(j)g_d(k-j) = \rho(k) \ast g_d(k) = y(k)\\
 %&+ \rho(1) (h(t-T)-h(t-2T))\\
 %&= \sum_{k=0}^{\infty} \rho(k)(h_k(t-kT)-h_k(t-(k+1)T))\\
 \end{align*}
-Verwendet wird neben der Bedingung \ref{ZOH:bed} die Faltungsumme für das ZOH-diskretisierten System, weshalb die Werte äquivalent zu $y(k)$ sind, der Antwort auf die Eingangsfolge $u(k)$. Für dieses spezielle, stufenförmige Eingangssignal liegen die Werte $y(k)$ also exakt auf dem kontinuierlichen Ausgangssignal:
+Verwendet wird neben der Bedingung \eqref{ZOH:bed} die Faltungsumme für das ZOH-diskretisierten System, weshalb die Werte äquivalent zu $y(k)$ sind, der Antwort auf die Eingangsfolge $u(k)$. Für dieses spezielle, stufenförmige Eingangssignal liegen die Werte $y(k)$ also exakt auf dem kontinuierlichen Ausgangssignal:
 \[
 y^*(t)|_{t=kT} =y(k) .
 \]
 
-\textbf{Berechung aus Übertragungsfunktonen.} Mit der Erkenntnis des letzten Abschnitts bedeutet ZOH-Diskretisierung, dass wir mit diskreten System exakt die kontinuierliche Antwort aus der Reihenschaltung eines Systems mit einem ZOH-Gleid (Halteglied erster Ordnung) treffen. Ein Impuls wird duch das ZOH-Glied zu einer stufenförmigen Anregung, diese ist Eingang von $G(s)$ und erzeugt unseren Ausgang $y^*(t)$, auf dem unsere diskreten Werte $y(k)$ liegen. Anders formuliert ist unser diskretes System damit die Impuls-Diskretisierung aus der Reihenschaltung von ZOH-Glied und System. Interpretiert man die diskreten Systemvariablen als Operatoren, könnte man schreiben
+\textbf{Berechnung aus Übertragungsfunktion des ZOH-Glieds.} Mit der Erkenntnis des letzten Abschnitts bedeutet ZOH-Diskretisierung, dass wir mit der Antwort des diskreten System exakt die kontinuierliche Antwort aus der Reihenschaltung eines Abtast-Halteglieds (erster Ordnung) mit dem System treffen. 
+Durch den Abtaster etsteht ein gewichteter Impulskamm.
+Ein Impuls wird durch das ZOH-Glied zu einer sprungförmigen Anregung, diese ist Eingang von $G(s)$ und erzeugt unseren Ausgang $y^*(t)$, auf dem unsere diskreten Werte $y(k)$ liegen. Anders formuliert ist unser diskretes System damit die Impuls-Diskretisierung aus der Reihenschaltung von ZOH-Glied und System. Interpretiert man die diskreten Systemvariablen als Operatoren, könnte man schreiben
 \begin{align*}
 G_{d,ZOH}\{ G(s) \} = G_{d,Impuls}\{ \frac{1-e^{-sT}}{s} \cdot G(s) \} ~~.
 \end{align*}
@@ -142,10 +176,42 @@ G_{d,ZOH}(z) =\frac{Tz}{(z-1)^2} - z^{-1}\cdot \frac{Tz}{(z-1)^2} =\frac{T(z-1)}
 }}
 
 \textbf{Nullstellen.}
-bei der Exact Discretization hat es $n-m$
-Zeros mehr wenn $D \neq 0$ ist, ansonsten $n-m-1$.
+Laut \cite{unbehauen2000RT2} kann nichts allgemeingültiges über die Lage der Nullstellen ausgesagt werde.
+Startet man mit der  Darstellung \eqref{eq:Impulsantwort}, dann folgt daraus für die Sprungantwort
+\[
+H(s) = \frac{1}{s} G(s) = \frac{d}{s} + \frac{1}{s}\sum_{i=0}^n \frac{b_i}{s-s_i}
+\]
+\begin{align}
+h(t) = d \sigma(t) + \sum_{i=0}^n -\frac{b_i}{s_i}(1- e^{s_i t}) \sigma(t)
+\end{align}
+bzw. abgetastet
+\[
+h(t) = d \sigma(kT) + \sum_{i=0}^n -\frac{b_i}{s_i}(1- e^{s_i kT}) \sigma(kT)~~.
+\]
+Z-Transformation liefert
+\[
+H(z) = d\frac{z}{z-1}  - \sum_{i=0}^n\frac{b_i}{s_i} \frac{ (1-e^{s_i T})z}{(z-1)(z-\underbrace{e^{s_i T}}_{z_i})}
+\]
+und
+\[
+G(z) = H(z) = \frac{z-1}{z}\cdot G(z)= d  + \sum_{i=0}^n\frac{b_i}{s_i} \frac{z_i-1}{z-{z_i}}~.
+\]
+Beim Aufmultiplizieren zur Ermittlung der Pol-/Nullstellenform von $G(z)$ erhält man hier ein Zählerpolynom der Ordnung
+\begin{itemize}
+\item $n$ für $d\neq 0$ und damit $n$ Nullstellen,
+\item $n-1$ für $d= 0$ und damit $n-$ Nullstellen.
+\end{itemize}
+
+
+\begin{figure}[H]
+\begin{center}
+  \includegraphics[width=0.8\textwidth]{bilder/zeitbereichsäquivalenzFOH}
+      \caption{Reihenschaltung aus kausalem FOH und System. Ausgang des FOH-diskretisierten Systems, verzögert um $T$, liegt auf kontinuierlichen Ausgang. %\citep{yawpitchroll2013}%
+             \label{fig:zeitbereichsäquivalenzFOH}}
+\end{center}
+\end{figure}
 
-\textbf{Zustandsraummodell.} In \cite{Lunze16b} wird ein diskretes System als Abtastsystem "`unter Beachtung der Eigenschaften des Abtasters und des Halteglieds"' aus dem kontinuierlichen ZRM hergeleitet. Dazu wird die Bewegungsgleichung für den stückweise stetigen (stufenförmigen) Eingang aufgeschrieben und dann abgetastet. In \cite{BINGULAC1992293} erhält man die fast identische Darstellung mit dem Hinweis darauf, dass es sich um eine ZOH-Diskretisierung handelt. Für regulär $\jv A_k$ entsteht das diskrete Zustandsraummodell \eqref{ZRM_disk} mit den Matrizen:
+\textbf{Zustandsraummodell.} In \cite{Lunze16b} wird ein diskretes System als Abtastsystem "`unter Beachtung der Eigenschaften des Abtasters und des Halteglieds"' aus dem kontinuierlichen ZRM hergeleitet. Dazu wird die Bewegungsgleichung für den stückweise stetigen (stufenförmigen) Eingang aufgeschrieben und dann abgetastet. In \cite{BINGULAC1992293} erhält man die fast identische Darstellung mit dem Hinweis darauf, dass es sich um eine ZOH-Diskretisierung handelt. Für reguläre $\jv A_k$ entsteht das diskrete Zustandsraummodell \eqref{ZRM_disk} mit den Matrizen:
 
 ZOH-Diskretisiertes ZRM:
 \begin{empheq}[box=\fbox]{align*}
diff --git a/5_Impulse-Diskretisierung.tex b/5_Impulse-Diskretisierung.tex
index 1f0becc..b1a0b09 100644
--- a/5_Impulse-Diskretisierung.tex
+++ b/5_Impulse-Diskretisierung.tex
@@ -39,8 +39,36 @@ Die theoretische Ermittlung korrespondierender diskreter Systeme aus der kontinu
 
 Die erste Zeile von Tabelle \ref{table:impuls} zeigt, wie der diskrete Integrator aus dem kontinuierlichen hervorgeht. Auf den Impuls antwortet der Integrator mit der Sprungfunktion (ohne Gewichtung durch die Zeitkonstante). Die dritte Spalte verzeichnet die Sprungfolge als deren Abtastung, also die diskretisierte Impulsantwort, welche in den Tabellen der Literatur oft nicht verzeichnet ist. Z-transformiert ergibt sich $G(z)$, in Zeile eins beispielsweise für den Integrator, auch wenn dieser gar nicht als numerischer Integrator verstanden werden kann (siehe fehlende Zeitkonstante). Außerdem hat dieser diskrete Integrator nicht die Bedeutung des Akkumulators bei den Verfahren nach Euler und Tustin. So gilt für Impulse-Diskretisierung nicht die Korrespondenz aus Spalte eins und vier als allgemeingültiges $s\leftrightarrow z$ Mapping, auch kann nicht einfach im Blockschaltbild der Differenzialgleichung jeder Integrator durch den Akkumulator ersetzt werden.
 
-Die vierte Zeile von Tabelle \ref{table:impuls} entspricht dem PT$_{1}$-Glied, womit die Einträge etwas abgewandelt sind im Vergleich zu typischen Korrespondenztabellen. $G(z)$ ist also das diskrete PT$_{1}$-Glied im Sinne einer Impuls-Diskretisierung.
+Die vierte Zeile von Tabelle \ref{table:impuls} entspricht dem PT$_{1}$-Glied, womit die Einträge etwas abgewandelt sind im Vergleich zu typischen Korrespondenztabellen. $G(z)$ ist also das diskrete PT$_{1}$-Glied im Sinne einer Impuls-Diskretisierung. Einige Beispiele sind hier in MATLAB\texttrademark~ gezeigt:
+
+\href{https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/Diskretisierung_Impulse.m}{File on Git: Diskretisierung\_Impulse.m}
+
+
 %\glqq{}{Boris ermöglicht die blockorientierte Simulation
  %        dynamischer Systeme nahezu beliebiger Art}\grqq{}
  %        \citep[S. 25]{boris2009}.
-         
+\textbf{Pol- und Nullstellen.} Um die Transformation bei der Impuls-Diskretisierung zu untersuchen wird in \cite{onlineSkript} die Darstellung
+\[
+G(s) = d + \sum_{i=0}^n \frac{b_i}{s-s_i}
+\]
+verwendet, wobei diese elementare Zerlegung in einen konstanten Anteil (Durchgriff) und eine Summe von reellen Systemen erster Ordnung nur für einfache, reelle Pole $s_i$ möglich ist. Konjugiert komplexe Polpaare kann man durch zwei komplexe Summanden zulassen, mehrfache Pole führen aber in der Partialbruchzerlegung auf Nennerpolynome höherer Ordnung. Trotzdem sei in dieser Form die Impulsantwort
+\begin{align}\label{eq:Impulsantwort}
+g(t) = d \rho(t) + \sum_{i=0}^n b_i e^{s_i t} \sigma(t)
+\end{align}
+bzw. abgetastet, wenn man den Durchgriff übernimmt, um den diskreten Impuls zu gewichten
+\[
+g(kT) = d \rho(K) + \sum_{i=0}^n b_i e^{s_i kT} \sigma(kT)~~.
+\]
+Z-Transformation liefert
+\[
+G(z) = d  + \sum_{i=0}^n b_i \frac{z}{z-e^{s_i T}}
+\]
+woraus sich erkennen lässt, wenn man auf Pol-/Nullstellenform aufmultiplizieren würde, dass
+\begin{enumerate}
+\item die Pole $z_i = e^{s_i T}$ heißen,
+\item nicht allgemein zu bestimmen ist, welche Nullstellen entstehen,
+\item ein Nennerpolynom vom Gerad $n$ entstehen würde und damit auch $n$ Nullstellen
+\item bei $d= 0$ eine der Nullstellen $z_{i0}=0$ heißt.
+\end{enumerate}
+
+
diff --git a/6_FOH-Diskretisierung.tex b/6_FOH-Diskretisierung.tex
index 5ffade1..900f0c1 100644
--- a/6_FOH-Diskretisierung.tex
+++ b/6_FOH-Diskretisierung.tex
@@ -1,25 +1,26 @@
 \subsection{FOH-Diskretisierung
          \label{sec:FOH-Diskretisierung}}
 
-Regt man ein System mit einer rampenförmigen Funktion $u(t)=m\cdot t$ an, dann wollen wir die Systemantwort bis auf die Steigung im Folgenden als Rampenantwort $r_k(t)$ bezeichnen.
+\textbf{Abtastung der Rampenantwort.} Regt man ein System mit einer rampenförmigen Funktion $u(t)=m\cdot t\cdot \sigma(t)$ an, dann wollen wir die Systemantwort bis auf die Steigung im Folgenden als Rampenantwort $r_k(t)$ bezeichnen.
 
 \begin{minipage}[c]{0.49\linewidth}
 \begin{align*}
-t = \int_0^t\sigma(\tau) d\tau
+\gamma(t)=\frac{u(t)}{m} = t\sigma(t) = \int_0^t\sigma(\tau) d\tau
 \end{align*}
 \end{minipage}
 \begin{minipage}[c]{0.49\linewidth}
 \begin{center}
 
 \begin{tikzpicture}%[domain=0:4]
-%\draw[very thin,color=gray] (-0.1,-1.1) grid (6,1.0);
-\draw[-] (-0.1, 1) node[left] {1} -- (0.1,1);
-%\draw[color=blue] (0.02, 0)  -- (0.02,1);
-\draw[color=blue] (0,0 )  -- (1,1);
-\draw[color=blue] (1,1 )  -- (2,0);
-\draw[-] (1, -0.1) node[below] {1} -- (1,0.1);
+\draw[-] (-0.1, 1.3) node[left] {$3T$} -- (0.1,1.3);
+%\draw[color=blue] (0.05,1 )  -- (2,1) node[right] {$u(t)$};
+\draw[-*] (0, -0.1) node[below] {} -- (0,0.1);
+\draw[-*] (0.4, -0.1) node[below] {$T$} -- (0.4,0.5);
+\draw[-*] (0.8, -0.1) node[below] {$$} -- (0.8,0.9);
+\draw[-*] (1.2, -0.1) node[below] {$3T$} -- (1.2,1.3);
+\draw (0, 0)  -- (1.3,1.3);
 \draw[->] (-0.2,0) -- (2.2,0) node[right] {$t$};
- \draw[->] (0,-0.5) -- (0,1.5) node[above right] {$g_{FOH}(t)$};
+ \draw[->] (0,-0.5) -- (0,1.5) node[above right] {$\gamma(t)=u(t); \gamma(k)$};
 \end{tikzpicture}
 
 \end{center}
@@ -30,73 +31,101 @@ so antwortet, dass die $y(k)=r_d(k)$ auf der kontinuierlichen Rampenantwort lieg
 \begin{align}\label{Ramp:bed}
 r_k(kT) =r_d (k)~~\laplace ~~R(z)~~
 \end{align}
-Um aus der z-Transformierten der diskreten Rampenantwort $R(z)$ auf die Übertragungsfunktion zu schließen, erinnern wir uns zunächst an den folgenden Zusammenhang
+
+\textbf{Ermittlung der Übertragungsfunktion.} Um aus der z-Transformierten der diskreten Rampenantwort $R(z)$ auf die Übertragungsfunktion zu schließen, erinnern wir uns zunächst an den folgenden Zusammenhang
 \begin{align*}%\label{Ramp:bed}
-h(k) = \sum_0^\infty g(k)
+h(k) = \sum_{k=0}^\infty g(k)
 \end{align*}
 basierend auf der Linarität des Systems, da die Sprungfunktion auch eine Summe von Impulsen darstellt. Daraus läst sich
 \begin{align*}%\label{Ramp:bed}
-H(z) = \sum_0^\infty z^{-k}G(z) = G(z) \sum_0^\infty z^{-k} =G(z) \frac{z}{z-1}
+H(z) = \sum_{k=0}^\infty z^{-k}G(z) = G(z) \sum_{k=0}^\infty z^{-k} =G(z) \frac{z}{z-1}
 \end{align*}
 herleiten.  
 
-Die diskrete Rampenfuntkion lautet wie folgt:
+Die diskrete Rampenfunktion lautet wie folgt:
+%\begin{minipage}[c]{0.49\linewidth}
+%\begin{align*}
+%\gamma (k) = kT = T\sum_{k=1}^\infty \sigma(k) = T(\sum_{k=0}^\infty \sigma(k) - \gamma(0))
+%\end{align*}
+%\end{minipage}
+%\begin{minipage}[c]{0.49\linewidth}
+%\begin{center}
+%
+%\begin{tikzpicture}%[domain=0:4]
+%%\draw[very thin,color=gray] (-0.1,-1.1) grid (6,1.0);
+%\draw[-] (-0.1, 1) node[left] {1} -- (0.1,1);
+%%\draw[color=blue] (0.02, 0)  -- (0.02,1);
+%\draw[color=blue] (0,0 )  -- (1,1);
+%\draw[color=blue] (1,1 )  -- (2,0);
+%\draw[-] (1, -0.1) node[below] {1} -- (1,0.1);
+%\draw[->] (-0.2,0) -- (2.2,0) node[right] {$t$};
+ %\draw[->] (0,-0.5) -- (0,1.5) node[above right] {$g_{FOH}(t)$};
+%\end{tikzpicture}
+%
+%\end{center}
+%\end{minipage}
+\begin{align}\label{eq:gammak}
+\gamma (k) = kT = T\sum_{j=1}^\infty \sigma(j) = T(- \gamma(k) + \sum_{j=0}^\infty \sigma(j) )
+\end{align}
+Die Summierung dieser Sprungfunktionen darf erst bei $j=1$ beginnen. Darauf antwortet das System mit
+
 \begin{minipage}[c]{0.49\linewidth}
 \begin{align*}
-\gamma (k) = kT = T\sum_{k=1} \infty \sigma(k) = T(\sum_{k=0} \infty \sigma(k) - 1)
+r(k) =  T\sum_{j=1}^\infty h(j) = T(- h(k) + \sum_{j=0}^\infty h(j) )
 \end{align*}
 \end{minipage}
 \begin{minipage}[c]{0.49\linewidth}
 \begin{center}
 
-\begin{tikzpicture}%[domain=0:4]
-%\draw[very thin,color=gray] (-0.1,-1.1) grid (6,1.0);
-\draw[-] (-0.1, 1) node[left] {1} -- (0.1,1);
-%\draw[color=blue] (0.02, 0)  -- (0.02,1);
-\draw[color=blue] (0,0 )  -- (1,1);
-\draw[color=blue] (1,1 )  -- (2,0);
-\draw[-] (1, -0.1) node[below] {1} -- (1,0.1);
-\draw[->] (-0.2,0) -- (2.2,0) node[right] {$t$};
- \draw[->] (0,-0.5) -- (0,1.5) node[above right] {$g_{FOH}(t)$};
+\begin{tikzpicture}[node distance=2.5cm,auto,>=latex', scale=3]
+    \node [int] (a) {$G_{d,FOH}(s)$};   %\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] {$$};
+    \node [coordinate] (end) [right of=c, node distance=2cm]{};
+    \path[->] (b) edge node {$\gamma(k)$} (a);
+    \path[->] (a) edge node {$r(k)$} (c);
 \end{tikzpicture}
 
 \end{center}
 \end{minipage}
-Die Summierung von Sprungfunktionen darf erst bei $k=1$ beginnen. Dementsprechend gilt
-\begin{align}
-R(z) =Tz^{-1} \frac{z}{z-1} H(z)= \frac{T}{z-1} H(z)
-\end{align}
 
 
+ Dementsprechend gilt für die z-transformierte Antwort
+\begin{align}
+R(z) &= T\sum_{k=1}^\infty H(z) z^{-k} = TH(z)  z^{-1}\sum_{k=0}^\infty z^{-k}\\ 
+&= Tz^{-1} \frac{z}{z-1} H(z)= \frac{T}{z-1} H(z)
+\end{align}
 
 Für ein System $G(s)$ lautet die Laplace-Transformierte der Rampenantwort
-\begin{align}\label{eq:Hvons}
-R(s) = \frac{1}{s^2} G(s) \Laplace r(t)~~,
+\begin{align}\label{eq:Rvons}
+R(s) = \frac{1}{s^2} G(s) ~\Laplace~ r(t)~~,
 \end{align}
 da sie aus der zweifachen Integration des Dirac-Inpulses entsteht.
-Eien Entsprechung im z-Bereich $R(z)$ kann eventuell einer Korrespondeztabelle entnommen bzw. mit deren Hilfe ermittelt werden. Über den Zusammenhang 
+Eine Entsprechung im z-Bereich $R(z)$ kann eventuell einer Korrespondeztabelle entnommen bzw. mit deren Hilfe ermittelt werden. Über den Zusammenhang 
 \[
 R(z) = \frac{T}{z-1}H(z)
 \]
-erhalten wir in unserem Fall
+erhalten wir dann
 \[
-G_{FOH}(z) = \frac{T}{z-1}  \frac{z}{z-1} R(z)~~.
+G_{FOH}(z) = \underbrace{\frac{T}{z-1}  \frac{z}{z-1}}_{\Laplace~ \gamma(k) \text{aus \eqref{eq:gammak}}} R(z)~~.
 \]
 
 
 \textit{\textbf{Beispiel: Die Übertragungsfunktion des Integrators lautet $G(s)=1/s$. Wir suchen nach der Z-Transformierten der Abtastung von $R(s)=1/s^3$. 
-Tabell \ref{table:impuls} liefert uns
+Tabelle \ref{table:impuls} liefert uns
 \[R(z)=\frac{T^2}{2}\frac{z(z+1)}{(z-1)^3}
 \]
 woraus wir
 \[
-G_{ZOH}(z) = R(z) \frac{z-1}{z}  \frac{z-1}{T} =\frac{T}{2}\frac{z+1}{z-1}
+G_{FOH}(z) = R(z) \frac{z-1}{z}  \frac{z-1}{T} =\frac{T}{2}\frac{z+1}{z-1}
 \]
 ermitteln.
 }
 }
 
-Interpretieren und berechnen kann man die FOH-Diskretisierung als Reihenschaltung eines nicht-kausalen FOH-Gliedes mit dem Originalsystem, welche dann Impuls-Diskretisiert wird. Die Antwort auf den Impuls bzw. eine Impulsfolge trifft das diskrete System exakt. Das Originalsystem wird derweil mit einer linearen Interpolation der Impulse, also mit einer Aneinanderreihung von rampenförmigen Signalen angeregt.  
+\textbf{FOH-Halteglied.} Interpretieren und berechnen kann man die FOH-Diskreti\-sierung als Reihenschaltung eines nicht-kausalen FOH-Gliedes mit dem Originalsystem, welche dann Impuls-Diskretisiert wird. Die Antwort auf den Impuls bzw. eine Impulsfolge trifft das diskrete System exakt. Das Originalsystem wird derweil mit einer linearen Interpolation der Impulse, also mit einer Aneinanderreihung von rampenförmigen Signalen angeregt.  
 Die Schreibweise
 \begin{align*}
 G_{d,FOH}\{ G(s) \} = G_{d,Impuls}\{ G_{FOH,sym}(s) \cdot G(s) \} ~~
@@ -125,16 +154,17 @@ G_{d,FOH}(z) =\frac{1}{T}\frac{T^2}{2} \frac{z(z+1)}{(z-1)^3}(z-2+z^{-1}) =\frac
 
 \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:
 
+ZRM-FOH:
 \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)\\
+ \jv A_d &=e^{\jv A_k T}         ~~&~~ \jv B_d &=\jv A_d(\jv A_d - \jv A_k T - \jv 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)
+  \jv C_d &=\jv C_k              ~~&~~ \jv D_d &= \jv C_d (\jv A_d - \jv A_k T - \jv 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:
-
+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 dort:
 \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
@@ -146,19 +176,22 @@ mit
 \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
+Das System \ref{ZRM_komisch} kann entsprechend Anhang A 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}
+\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
+\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).
 
+\href{https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/Diskretisierung_FOH.m}{File on Git: Diskretisierung\_FOH.m}
+
+
 
 
 %\begin{eqnarray}
diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf
index 23ac2bb435a96118cebb9d995af66d65ef1b242f..4ea3b4f80b4fe654fb5316c4d0d76898e5794eb4 100644
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diff --git a/Regelungstechnik.tex b/Regelungstechnik.tex
index e69de29..1011067 100644
--- a/Regelungstechnik.tex
+++ b/Regelungstechnik.tex
@@ -0,0 +1,72 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{PID und PID-T1-Regler
+         \label{sec:PID}}
+				
+\textbf{Kontinuierlicher Regler.} Regler ermitteln in der Regel die Stellgröße, weshalb $u(t)$ jetzt Ausgangssignal des Reglers ist, in Abhängigkeit vom Regelfehler $e(t)$. Daraus folgt für den PID-Regler die Darstellung
+\begin{align}
+u_{PID}(t) = k_p \left(e(t)+\frac{1}{T_I} \int_0^t e(\tau) d \tau + T_D \dot e(t)\right)
+\end{align}
+\begin{align}
+G_{PID}(s) = \frac{U(s)}{E(s)} = k_p(1+\frac{1}{T_Is}+T_D s)
+\end{align}
+Anstelle des nicht-kausalen Differenzierers wird auch die Reihenschaltung aus Differenzierer und Verzögerungsglied erster Ordnung (PT1) verwendet  
+\begin{align}
+G_{PI-DT_1}(s) = \frac{U(s)}{E(s)} = k_p(1+\frac{1}{T_Is}+\frac{T_Ds}{1+T_Vs})~~,
+\end{align}
+womit ein vierter Parameter $T_V$ festzulegen ist. Die Anteile P, I und D (bzw. DT1) können im diskreten wie im kontinuierlichen unabhängig voneinander in einer Parallelschaltung realisiert werden. Ändert sich die Führungsgröße $w(t)$ möglicherweise sprunghaft, z. B. durch manuelle Vorgabe, so bietet es sich an, diese nicht über den D-Anteil zu führen. In der parallelen Darstellung der Anteile lässt sich dies leicht berücksichtigen
+\begin{align}
+u_{PID*}(t) = k_p \left(e(t)+\frac{1}{T_I} \int_0^t e(\tau) d \tau + T_D  (- \dot y_m(t)\right)~~,
+\end{align} 
+so dass hier nur die Messgröße über den D-Anteil geführt wird, siehe Bild \ref{fig:PIDmod}.
+
+\begin{figure}[H]
+\begin{center}
+  \includegraphics[width=0.8\textwidth]{bilder/PIDmod}
+      \caption{Modifizierter PID-Regler. %\citep{yawpitchroll2013}%
+             \label{fig:PIDmod}}
+\end{center}
+\end{figure}
+
+
+\textbf{Diskretisierter Regler.} Unterschiedliche Diskretisierungsansätze führen auch bei identischer Struktur auf verschiedene Darstellungen des diskreten PID-Reglers in der Literatur. Schlägt man in \cite{lutz:2005} nach, findet man die Angabe zweier Typen von PID-Reglern, welche sich in der Integration unterscheiden:
+\begin{align}
+\text{Typ I:}~~ &G_I(z)=\frac{k_p}{T_I}\frac{T}{z-1}\\
+\text{Typ II:}~~ &G_I(z)=\frac{k_p}{T_I}\frac{Tz}{z-1} ~\Laplace~ u^I_k = u^I_{k-1} + k_p\frac{T}{T_I} e_k 
+\end{align} 
+Die verzögerte  Integration des Typ I entspricht der Integration bei Forward-Euler-Diskretisierung. Ob diese hier geeignet ist, ist fraglich und der Fokus liegt auf der Variante Typ II liegen, welche der schnelleren Integration bei Backward-Euler-Diskretisierung entspricht. Der D-Anteil wird in beiden Fällen über den Differenzenquotienten
+\begin{align}
+u_k^D = k_P T_d\frac{e_k-e_{k-1}}{T}\\
+\end{align} 
+gebildet (entsprechend Backward-Euler-Diskretisierung) und mit dem P-Anteil komplettiert:
+\begin{align}
+u_k^P = k_P e_k\\
+\end{align} 
+womit sich der angegebene Regler zu
+\begin{align}
+u_k &=u_k^P + u_k^I + u_k^D\\
+ &=k_p(e_k + T_d\frac{e_k-e_{k-1}}{T} +\frac{T}{T_I}e_k)  + \underbrace{u^I_{k-1}}_{=u_{k-1}-u^P_{k-1}-u^D_{k-1}}\\
+&= u_{k-1} + k_p\left([1 +\frac{T}{T_I} + \frac{T_D}{T}]e_k -[1+2\frac{T_D}{T}]e_{k-1} + \frac{T_D}{T}e_{k-2}\right)
+\end{align}
+Tabelliert erhält man in \cite{lutz:2005} für beide Typen die Koeffizienten der Übertragungsfunktion für den \textit{Stellungsalgorithmus} des PID-Reglers sowie der Spezialfälle.
+
+In \cite{Lunze16b} und \cite{unbehauen2000RT2} wird jeweils vorgeschlagen, den I-Anteil durch die Trapez-Regel entsprechend Tustin-Diskretisierung zu realisieren, also für den echten PID-Regler 
+\[
+U(z) = kp\left(1 + \frac{T}{2T_I} \frac{z+1}{z-1} +\frac{k_D}{T} \frac{z-1}{z} E(z) \right)
+\]
+In  \cite{unbehauen2000RT2} wird der PID-T1-Regler angegeben. Dabei sei es von Vorteil, das PT1 Glied mit der Entsprechung
+...
+zu diskretisieren.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Diskretisierung und Implementierung allgemeiner Regler
+         \label{sec:allgemeineRegler}}
+				
+				
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Abtastsystem, (ZOH-)Diskretisierung der Strecke 
+         \label{sec:Abtastsystem}}
+				
+				
+
+
+ 
\ No newline at end of file
diff --git a/anhang.tex b/anhang.tex
index 065de6a..a557dbb 100644
--- a/anhang.tex
+++ b/anhang.tex
@@ -65,25 +65,29 @@ um den Entwicklungspunkt $x_0$ wird als bekannt vorausgesetzt. Angewendet auf de
 \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
+stellt sie den Ausgangspunkt dar für z. B. in \cite{wiki1} hergeleitete, variierende Darstellungen von Entwicklungen. Für die gesuchte Reihe 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\left(\frac{2x}{2}\right) = \ln\left( \frac{1+\frac{x-1}{x+1}   }{1- \frac{x-1}{x+1}  }\right) &=  \ln\left(1+\frac{x-1}{x+1}\right) &- \ln\left(1-\frac{x-1}{x+1}\right)\\
-&= \ln(1+y) &- \ln(1-y)
+\ln(x) = \ln\left(\frac{2x}{2}\right) = \ln\left( \frac{1+\frac{x-1}{x+1}   }{1- \frac{x-1}{x+1}  }\right) &=  \ln\left(1+\frac{x-1}{x+1}\right) - \ln\left(1-\frac{x-1}{x+1}\right)\\
+&= \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}\\
+&= 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
+und mit
+\[
+y =\frac{x-1}{x+1}
+\] 
+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}\\
+\ln(x) &= 2\sum_{k=1}^{\infty} \frac{1}{2k-1} \left(\frac{x-1}{x+1}\right)^{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*}
 
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diff --git "a/bilder/zeitbereichs\303\244quivalenzFOH.PNG" "b/bilder/zeitbereichs\303\244quivalenzFOH.PNG"
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diff --git "a/bilder/zeitbereichs\303\244quivalenzZOH.PNG" "b/bilder/zeitbereichs\303\244quivalenzZOH.PNG"
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HcmV?d00001

diff --git a/hauptdatei.bib b/hauptdatei.bib
index 0af2515..d0b7d28 100644
--- a/hauptdatei.bib
+++ b/hauptdatei.bib
@@ -55,6 +55,14 @@ year = {2021},
 note = {\url{https://www.eit.hs-karlsruhe.de/mesysto/fileadmin/downloads/teilB/PDFs/Musterloesungen/Skript_TDS_Musterloesungen_Kapitel_6.pdf}}
 }
 
+@MISC{onlineSkript,
+author = {Manfred Strohrmann},
+title = {{Systemtheorie Online}},
+month = {März},
+year = {2015},
+note = {\url{https://www.eit.hs-karlsruhe.de/mesysto/nc/downloads/downloads-teil-a-zeitkontinuierliche-signale-und-systeme/downloadstatistik.html?download=Skript_SYS.pdf&did=13}}
+}
+
 @article{BINGULAC1992293,
 title = {Identification of First-Order Hold Continuous-Time Systems},
 journal = {IFAC Proceedings Volumes},
@@ -86,4 +94,17 @@ title = {Natural logarithm},
 month = {März},
 year = {2021},
 note = {\url{https://en.wikipedia.org/wiki/Natural_logarithm}}
-}
\ No newline at end of file
+}
+
+@book{lutz:2005,
+  added-at = {2006-07-14T23:24:19.000+0200},
+  author = {Lutz, H. and Wendt, W.},
+  biburl = {https://www.bibsonomy.org/bibtex/29237f4d8a2eb803dd6947c7ded3a6e68/mobst},
+  interhash = {8e1c781aa6168842a12288c197868033},
+  intrahash = {9237f4d8a2eb803dd6947c7ded3a6e68},
+  keywords = {Elektrotechnik},
+  publisher = {H. Deutsch},
+  timestamp = {2006-07-14T23:24:19.000+0200},
+  title = {{Taschenbuch der Regelungstechnik}},
+  year = 2005
+}
-- 
GitLab