From 36d8f51f46d4f4f26010ef7ada7c891cafd5fe33 Mon Sep 17 00:00:00 2001
From: Jan Falkenhain <jan.falkenhain@hs-bochum.de>
Date: Fri, 30 Apr 2021 14:50:28 +0200
Subject: [PATCH] 13erd commit

---
 1_Einleitung.tex              |  12 +++---
 2_Grundlagen.tex              |  32 ++++++++++----
 3_Euler-Diskretisierung.tex   |  17 +++++---
 4_ZOH-Diskretisierung.tex     |  28 ++++++------
 5_Impulse-Diskretisierung.tex |  56 +++++++++++++++++++++---
 6_FOH-Diskretisierung.tex     |  79 ++--------------------------------
 7_Tustin-Diskretisierung.tex  |  23 +++++++---
 8_Matched-Filter.tex          |   4 +-
 FB_Diskretisierung.pdf        | Bin 2390348 -> 2403261 bytes
 Regelungstechnik.tex          |  31 ++++++-------
 anhang.tex                    |   8 ++--
 hauptdatei.bib                |   8 ++++
 12 files changed, 152 insertions(+), 146 deletions(-)

diff --git a/1_Einleitung.tex b/1_Einleitung.tex
index 3bb6824..7f4c7c4 100644
--- a/1_Einleitung.tex
+++ b/1_Einleitung.tex
@@ -10,15 +10,15 @@
 
 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 kontinuierlichen Modells in Reihe geschaltet mit einem Halteglied erster Ordnung (ZOH, Zero-Order-Hold) nachbilden will. Das resultierende System heißt Abtastsystem, worauf in Kapitel \ref{sec:Regelung} eingegangen wird. Man erhält es durch ZOH-Diskretisierung (Kap. \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 (Kap. \ref{sec:Tustin-Approximation})  oder Matched-Filter-Diskretisierung (Kap. \ref{sec:FOH-Diskretisierung}) verwendet,
 %oder FOH-Diskretisierung (Kap. \ref{sec:FOH-Diskretisierung}) verwendet
-welche auf "`schnellere"' diskrete Systeme führen, was auch für FOH-Diskretisierung (Kap. \ref{sec:FOH-Diskretisierung}) gelten müsste.
+welche auf "`schnellere"' diskrete Systeme führen, was auch für FOH-Diskretisierung (Kap. \ref{sec:FOH-Diskretisierung}) gelten müsste. Der gesamte Bereich der Diskretisierung in der Regelungstechnik wird in der Videoserie \cite{BrianDouglas} sehr geeignet dargestellt.
 
-In der digitalen Signalverarbeitung approximiert man anlaloge 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 ab (Kap. \ref{sec:Impulse-Diskretisierung}). Auch der Matched-Filter Ansatz, bei welchem Pole und Nullstellen des diskreten Systems gezielt gesetzt werden, spielt in der digitalen Signalverarbeitung eine größere Rolle.
+In der digitalen Signalverarbeitung approximiert man analoge 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 ab (Kap. \ref{sec:Impulse-Diskretisierung}). Auch der Matched-Filter Ansatz, bei welchem Pole und Nullstellen des diskreten Systems gezielt gesetzt werden, spielt in der digitalen Signalverarbeitung eine größere Rolle.
 
 Weitere Motive zur Ermittlung diskreter Systeme können in der Analyse, Simulation, oder modellbasierten Diagnose technischer Systeme liege. Die Ermittlung eines Modells aus Messungen an einem realen System wird als Identifikation bezeichnet. So kann z. B. näherungsweise eine Impulsantwort gemessen werden oder der Eingang als stückweise konstant (ZOH-Modell) oder als lineare interpolierende Rampe (FOH-Modell) interpretiert werden. Ein diskretes Modell aus den Messwerten zu ermitteln, 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{sec:Grundlagen} sollen in Kapitel \ref{sec:Diskretisierungsmethoden} alle genannten Ansätze auch aus verschiedenen Blickwinkeln betrachtet werden, um einen systematischen Überblick zu ermöglichen.
 
-Die Ansätze können, für die hier behandelten lineare Systeme, sowohl im Zeit- als auch im Frequenzbereich betrachtet werden. Generell ist von Interesse, wie letztlich jede Methode die Integration im diskreten numerisch umsetzt. Nur einige Methoden lassen sich durch Ersetzen aller Integratoren durch diskrete Akkumulatoren umsetzten. Damit ist gemeint, dass in der jeweiligen kontinuierlichen Darstellung der Differenzialgleichung jder Integrator durch den Akkumulator ersetzt wird.  Ob ein stabiles kontinuierliches System in ein stabiles diskretes transformiert wird, hängt von der Abbildung der Polstellen ab. Das diskrete System hat immer die selbe Anzahl an Polstellen. Für dynamische Eigenschaften sind aber auch die Nullstellen relevant. Diese können abgebildet werden oder neu bei der Diskretisierung  entstehen. Wichtige Eigenschaften werden in Tabelle \ref{table:massen} aufgeführt. Ihre Bedeutung soll in den einzelnen Kapitel näher erläutert werden. Neben der Übertragungsfunktion (als Transformationspaar mit der z-Übertragungsfunktion) sollen auch Zustandsraummodelle betrachtet werden.  
+Die Ansätze können, für die hier behandelten lineare Systeme, sowohl im Zeit- als auch im Frequenzbereich betrachtet werden. Generell ist von Interesse, wie letztlich jede Methode die Integration im diskreten numerisch umsetzt. Nur einige Methoden lassen sich durch Ersetzen aller Integratoren durch diskrete Akkumulatoren umsetzten. Damit ist gemeint, dass in der jeweiligen kontinuierlichen Darstellung der Differenzialgleichung jeder Integrator durch den Akkumulator ersetzt wird.  Ob ein stabiles kontinuierliches System in ein stabiles diskretes transformiert wird, hängt von der Abbildung der Polstellen ab. Das diskrete System hat immer die selbe Anzahl an Polstellen. Für dynamische Eigenschaften sind aber auch die Nullstellen relevant. Diese können abgebildet werden oder neu bei der Diskretisierung  entstehen. Wichtige Eigenschaften werden in Tabelle \ref{table:massen} aufgeführt. Ihre Bedeutung soll in den einzelnen Kapitel näher erläutert werden. Neben der Übertragungsfunktion (als Transformationspaar mit der z-Übertragungsfunktion) sollen auch Zustandsraummodelle betrachtet werden.  
 
 \begin{landscape}
 
@@ -88,8 +88,8 @@ Damit lassen sich Systeme erzeugen, Darstellungen ineinander überführen und di
 
 \href{https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/Diskretisierung_mit_MATLAB.m}{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. 
+Hier wird ein System zweiter Ordnung (ohne Durchgriff, mit komplexen Eigenwerten) diskretisiert nach den Methoden ZOH (Zero-Order Hold), FOH (First-Order Hold), Impuls 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 konzipiert 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 Impuls-approximierte Sprungantwort nicht gilt, kann zu dem Schluss führen, dass Impuls-Approximation wenig geeignet ist, das kontinuierliche System zu approximieren. 
 
 \begin{figure}[H]
   \begin{center}
@@ -101,7 +101,7 @@ Der Verlauf für das ZOH-approximierte System liegt zu den Zeitpunkten $kT$ exak
 
 Aus einer anderen Perspektive wird aber lediglich ein anderer Aspekt, nämlich die Impulsantwort exakt nachgebildet, bzw. bei einer Identifikation aus Messdaten "`möglichst gut"' approximiert. Ein Vergleich der diskreten mit der kontinuierliche Impulsantwort zeigt Bild \ref{fig:Impulseantwort}.
 
-\textbf{Digitale Regelung.} Für die digitale Regelung ist zunächst ein Verständnis der Bedeutung der Abtastzeit fundamental. Klar ist, dass eine konstante Abtastzeit $T$ kleiner als die dominanten Zeitkonstanten des zu regelnden Prozesses sein muss. Der Regler muss schneller sein als der Prozess. Zur Realisierung benötigt man eine echtzeitfähige Hardware, welche sicherstellt, dass $T$ bis auf tolerierbare Schwankungen (Jitter) eingehalten wird. Eine zu langsame Hardware ist prinzipiell nicht einsetztbar, während die Echtzeitfähigkeit auch fordert, dass der Regel-Algorithmus nicht in zu kurzen Abständen aufgerufen wird.
+\textbf{Digitale Regelung.} Für die digitale Regelung ist zunächst ein Verständnis der Bedeutung der Abtastzeit fundamental. Klar ist, dass eine konstante Abtastzeit $T$ kleiner als die dominanten Zeitkonstanten des zu regelnden Prozesses sein muss. Der Regler muss schneller sein als der Prozess. Zur Realisierung benötigt man eine echtzeitfähige Hardware, welche sicherstellt, dass $T$ bis auf tolerierbare Schwankungen (Jitter) eingehalten wird. Eine zu langsame Hardware ist prinzipiell nicht einsetzbar, während die Echtzeitfähigkeit auch fordert, dass der Regel-Algorithmus nicht in zu kurzen Abständen aufgerufen wird.
 Echtzeitfähige Hardware kann z. B. durch einen priorisierten Interrupt realisiert werden, welcher mittels eines Timers (Uhr) mit der Periode $T$ durchgeführt wird, um den Regel-Algorithmus auszuführen. Der Algorithmus muss natürlich in der Zeit $T$ ausführbar sein, was bei den hier ermittelten diskreten Filtern mit konstanter Anzahl von Operationen leicht sichergestellt werden kann. Aufwendige Algorithmen wie numerische Optimierungen in der modellprädiktiven Regelung erfordern eine genauere Betrachtung.
 
 Kontinuierliche Reglergesetze lassen sich mit unterschiedlichen Ansätzen diskretisieren, wobei das resultierende diskrete Reglergesetz von der Abtastzeit abhängt. Dies soll am Beispiel des PID- bzw. PID-T1-Reglers gezeigt werden, für welchen in der Literatur verschiedene Differenzengleichungen angegeben werden. Genauso ergeben sich je nach Diskretisierungsmethode für allgemeine, dynamische Regler verschiedene diskrete Reglergesetze. Aus der z-Übertragungsfunktion können die Parameter einer kanonischen Filterstruktur für den Regler direkt abgelesen werden. 
diff --git a/2_Grundlagen.tex b/2_Grundlagen.tex
index 3f33971..415d692 100644
--- a/2_Grundlagen.tex
+++ b/2_Grundlagen.tex
@@ -49,7 +49,7 @@ Das Ausgangssignal $y(t)$ ergibt sich durch die Faltung (Faltungsintegral) der G
 \[
 y(t) = u(t)\ast g(t) = \int_{-\infty}^t u(\tau) g(t-\tau) d\tau~~.
 \]
-Die Übertragungsfuktion als Bruch zweier Polynome in $s$
+Die Übertragungsfunktion als Bruch zweier Polynome in $s$
 \[
 G(s) = \frac{Y(s)}{U(s)} = K_k\frac{\Pi_{i=1}^q (s-s_{0i})}{\Pi_{i=1}^n (s-s_i)}
 \]
@@ -145,8 +145,17 @@ Die Abbildung $z=e^{sT}$ aus der $s$-Ebene in die $z$-Ebene ist in Bild \ref{fig
  \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 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
+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 Lösung des Systems kann allgemein mit
+\begin{align}\label{ZRM_kontLsg}
+\jv y(t) = \jv C_k e^{\jv A_kt}\jv x_0 + \jv C_k\int_0^t e^{\jv A_k(t-\tau)}\jv B_k \jv u(\tau) d\tau  + \jv D_k \jv u(t)
+\end{align}
+angegeben werden.
+Darin stellt $\e^{\jv A t}$ eine Matrix-Exponentialfunktion dar: 
+\begin{align}\label{eq:MEF}
+e^{\jv A t} = \sum_{i=0}^{\infty} \frac{\jv A^i t^i}{i!} %\text{MATLAB\texttrademark: \verb! expm(A)!}
+\end{align}
+
+Die Laplace-Transformierte der Zustandsgleichung \eqref{ZRM_kont} lautet
 \begin{align*}
  s{\jv X}(s) &= \jv A \jv X(s) + \jv B \jv U(s)~~,
 \end{align*}
@@ -155,14 +164,19 @@ Aufgelöst nach $\jv X(s)$ und eingesetzt in die transformiert Ausgabegleichung
 \begin{align*}
  {\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 $s=0$ (bzw. $\dot{\jv x}(t)=0$) ergibt sich die statische Verstärkung $K_s$, bzw, hier als Matrix $\jv K_s$.
+Für $s=0$ (bzw. $\dot{\jv x}(t)=0$) ergibt sich die statische Verstärkung $K_s$, bzw. hier als Matrix $\jv K_s$.
 
 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 y_k &= \jv C_d \jv x_k + \jv D_d \jv u_k\nonumber
 \end{align}   
-ein lineares, diskretes ZRM mit Zustandsübergangsmatrix $\jv A_d$ in der Zustandsgleichung und entsprechenden Ein- und Ausgangsmatrizen $\jv B_d$ und $\jv C_d$. 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. $G_{2,3}(s)=Y_3(s)/U_2(s)$).
+ein lineares, diskretes ZRM mit Zustandsübergangsmatrix $\jv A_d$ in der Zustandsgleichung und entsprechenden Ein- und Ausgangsmatrizen $\jv B_d$ und $\jv C_d$. Die Auswertung erfordert immer die Kenntnis des Anfangszustands $\jv x_0$. Die Lösung lautet
+\begin{align}\label{ZRM_diskLsg}
+\jv y_k = \jv C_d \jv A_d^k\jv x_0 + \jv C_d\sum_{i=0}^{k-1}\jv A_d^{k-1-i}\jv B_d \jv u_i + \jv D_d \jv u_k~~.
+\end{align}
+
+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*}
@@ -193,11 +207,11 @@ nach $\jv X(z)$ aufgelöst und in die Ausgabegleichung eingesetzt werden, um die
 \end{align*}
 zu ermitteln. Letztere unterscheidet sich formal, da im diskreten Fall der statische Zustand für $z=e^{0\cdot T}=1$ oder im Zeitbereich mit dem Ansatz $\jv x(k+1) = \jv x(k)$ ermittelt wird. 
 
-Zur Klasse der linearen Systeme gehören auch Totzeitsystem, welche Verzögerungen um die Totzeit $T_t$ erfassen.
+\textbf{Totzeiten.} Zur Klasse der linearen Systeme gehören auch Totzeitsystem, welche Verzögerungen um die Totzeit $T_t$ erfassen.
 \begin{align*}
 	y(t) = u(t-T_t)
 \end{align*}
-Im der Übertragungsfunktion wird der gebrochen rationale Teil um Faktoren
+In der Übertragungsfunktion wird der gebrochen rationale Teil um Faktoren
 \begin{align*}
  G_t(s)=e^{-sT_t}
 \end{align*}
@@ -205,7 +219,7 @@ ergänzt. Wenn die Totzeit ein Vielfaches der Abtastzeit $T_t= m\cdot T$ beträg
 \begin{verbatim}
 sys = thiran(tau, Ts) % tau =T_t; Ts ist die Abtastzeit
 \end{verbatim}
-Auch wenn Systemen Einagangs- und Ausgangsverzögerungen (Totzeiten) hinzugefügt werden und mit beliebigem Verfahren diskretisiert werden, kommt Thiran-Approximation automatisch zum Einsatz.
+Auch wenn Systemen Eingangs- und Ausgangsverzögerungen (Totzeiten) hinzugefügt werden und mit beliebigem Verfahren diskretisiert werden, kommt Thiran-Approximation automatisch zum Einsatz.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Halteglieder (ZOH und FOH) in Theorie und Praxis%
@@ -302,7 +316,7 @@ 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*}
-Mögliche Realisierungenin MATLAB/Simulink\texttrademark~zeigt die folgende Datei:
+Mögliche Realisierungen in 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}
 
diff --git a/3_Euler-Diskretisierung.tex b/3_Euler-Diskretisierung.tex
index eb355a4..fa7934c 100644
--- a/3_Euler-Diskretisierung.tex
+++ b/3_Euler-Diskretisierung.tex
@@ -17,11 +17,14 @@ Die Differenzengleichung des diskreten Differenziators mit Eingang $u(k)$ lautet
 \begin{align*}
 y(k)&= \frac{u(k+1)-u(k)}{T}~~.
 \end{align*}
-Sie ist nicht realisierbar, weil nicht kausal. Allerdings gilt ja auch der ideale kontinuierliche Differenziator als nicht-kausal. Die Übertragunsfunktion ergibt sich durch z-Transformation zu
+Sie ist nicht realisierbar, weil nicht kausal. Allerdings gilt ja auch der ideale kontinuierliche Differenziator als nicht-kausal. Die Übertragungsfunktion ergibt sich durch z-Transformation zu
 \begin{align*}
 Y(z) = \frac{zU(z)- U(z)}{T}  ~~\Leftrightarrow~~ \frac{Y(z)}{U(z)}=\frac{z-1}{T}~\leftrightarrow G(s)=s
 \end{align*}
-und führt auf den Zusammenhang $sT+1 \equiv z$.
+und führt auf den Zusammenhang $sT+1 \equiv z$. Dies entspricht auch der Taylorreihenentwicklung \eqref{eq:Taylorreihe} für $f(s)=e^{sT}$ um $s_0=0$ bis zum Term erster Ordnung:
+\begin{align}\label{eq:TaylorE}
+z = e^{sT} = \sum_{n=0}^\infty \frac{T^n}{n!} s^n = 1+ sT + \frac{(sT)^2}{2} \hdots \approx 1+sT
+\end{align}
 
 \textbf{Integration.} Bei der Euler-Diskretisierung ersetzt man jede elementare Differenziation entsprechend, bzw. jede Integration, welche z. B. im Signalflussgraphen der Differenzialgleichungen auftritt. Als Umkehroperation zur Bildung des Differenzenquotienten leiten wir die diskrete Integration bzw. Akkumulation wie ab. Wir vertauschen Ein- und Ausgang und bilden den Differenzenquotienten des Ausgangssignals: 
 \[u(k)= \frac{y(k+1)-y(k)}{T}\]
@@ -97,7 +100,7 @@ Eine Eigenschaft des Ansatzes ist damit, dass eine eindeutige $s-z$-Transformati
 \end{figure}
 
 Betrachtet man die Verschiebung der Pole der linken s-Halbebene, so können stabile Systeme in instabile diskrete Systeme transformiert werden. 
-Man kann sich vorstellen, dass nur in Ausnahmefällen stabile Pole des kontinuierlichen Systems im Einheitskreis der z-Ebene landen und somit auch stabile Pole des diskreten Systems darstellen. Das dargestellte, grenzstabile Polpaar wird auf instabile Pole des diskreten Systes führen, aber auch zu weit links in der s-Ebene liegende Pole transformieren bei zu geringer Abtastrate möglicherweise nicht auf stabile Pole.
+Man kann sich vorstellen, dass nur in Ausnahmefällen stabile Pole des kontinuierlichen Systems im Einheitskreis der z-Ebene landen und somit auch stabile Pole des diskreten Systems darstellen. Das dargestellte, grenzstabile Polpaar wird auf instabile Pole des diskreten Systems führen, aber auch zu weit links in der s-Ebene liegende Pole transformieren bei zu geringer Abtastrate möglicherweise nicht auf stabile Pole.
 
 
 \textbf{Zustandsraummodell.} Die Art der Polverschiebung zeigt sich auch bei Diskretisierung der Zustandsgleichung des ZRM \eqref{ZRM_kont}, wo mit dem Differenzenquotienten \eqref{Forward_diffq} folgt
@@ -128,7 +131,7 @@ ZRM-EF:
 
 \textbf{Beispiel.} Die Sprunganwort (Bild \ref{fig:step-ForwardEuler}) für das EF-diskretisierte Originalsystem 
 \[G(s)=\frac{s+1}{s^2+s+1}\] 
-liegt für $T=0.5$ im Vergleich zu anderen Verfahren (Bild \ref{fig:Sprungantwort}) weiter von der kontinuierlichen Sprungantwort weg. Wie Bild \ref{fig:step-ForwardEuler} zeigt, nähern sich die diskreten Werte dem kontinuierlichen Original bei Reduktion der Abtastzeit. In der Simulationstechnik wird genau die notwendige, sehr geringe Schrittweite als Ausschlusskriterium für das Verfahren genannt. Für größere $T$ wird das diskrete System instabil. Die statische Verstärkung des Originalsystems $h(t\rightarrow\infty)$ entspricht unabbhängig von $T$ der, des diskreten Systems. Das dies im Allgemeinen der Fall ist, kann anhand der Darstellungen in ZRM-Form gezeigt werden:
+liegt für $T=0.5$ im Vergleich zu anderen Verfahren (Bild \ref{fig:Sprungantwort}) weiter von der kontinuierlichen Sprungantwort weg. Wie Bild \ref{fig:step-ForwardEuler} zeigt, nähern sich die diskreten Werte dem kontinuierlichen Original bei Reduktion der Abtastzeit. In der Simulationstechnik wird genau die notwendige, sehr geringe Schrittweite als Ausschlusskriterium für das Verfahren genannt. Für größere $T$ wird das diskrete System instabil. Die statische Verstärkung des Originalsystems $h(t\rightarrow\infty)$ entspricht unabhängig von $T$ der, des diskreten Systems. Das dies im Allgemeinen der Fall ist, kann anhand der Darstellungen in ZRM-Form gezeigt werden:
 \begin{align*}
 G_k(s=0) = \jv D- \jv C\jv A_k^{-1} \jv B_k &\stackrel{?}{=}  \jv D+ \jv C(\jv I-\jv A_d)^{-1} \jv B_d =G_d(z=1)\\
 \Leftrightarrow~~ - \jv A_k^{-1} \jv B_k &{=}  \jv  (\jv I-\jv A_d)^{-1} \jv B_d\\
@@ -139,7 +142,7 @@ G_k(s=0) = \jv D- \jv C\jv A_k^{-1} \jv B_k &\stackrel{?}{=}  \jv D+ \jv C(\jv I
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \subsubsection{Euler-Backward-Diskretisierung (implizit)
-            \label{sec:Euler-Forward-Diskretisierung}}
+            \label{sec:Euler-Backward-Diskretisierung}}
 
 \textbf{Differenzenquotient.} Der Ansatz Euler-Backward (EB) bedeutet, einen kausalen diskreten Differenziator zu bilden, bei dem die Zeitableitung durch den Differenzenquozienten aus dem aktuellen und dem \textbf{vergangenen} Vorgängerwert des Signals ersetzt wird:
 \begin{align}\label{eq:firstOrderDiff}
@@ -249,7 +252,7 @@ K\frac{\Pi_{i=1}^q (-s_{0i}T)}{\Pi_{i=1}^n (-s_iT)}\cdot (T)^{n-q} = G_d(1) \che
 \end{align}
 
 \begin{itemize}
-\item Die Umformung von Gl. \eqref{ZRM:kompliziert} in ein ZRM ist etwas aufwendiger, weil hier die vermeindliche Zustandsgröße $\jv x(k+1)$ vom Eingang $\jv u(k+1)$ zum gleichen Zeitpunkt abhängt. Mit einem modifizierten Zustand sollte die Darstellung gelingen (siehe Anhang A und Fortsetzung). 
+\item Die Umformung von Gl. \eqref{ZRM:kompliziert} in ein ZRM ist etwas aufwendiger, weil hier die vermeidliche Zustandsgröße $\jv x(k+1)$ vom Eingang $\jv u(k+1)$ zum gleichen Zeitpunkt abhängt. Mit einem modifizierten Zustand sollte die Darstellung gelingen (siehe Anhang A und Fortsetzung). 
 \item Es lässt sich aber erkennen, dass die Zustandsübergangsmatrix $\jv A_d= (\jv I-T\jv A_k)^{-1}$ lautet.
 \item Plausibilisiert man sich die Eigenwerttransformation anhand der stabilen Eigenwerte $\lambda_i(jv A_k)<0$ auf der reellen Achse, so kann das mit der speziellen Form der Diagonalmatrix für $\jv A_k$ gelingen. Es gilt dann $\hat{\lambda}_i(\jv I-T\jv A_k)>1$  und damit für die inverse Matrix $0<\lambda_i(\jv A_d)<1$, also im angedeuteten Bildbereich auf der positiven reellen Achse im Einheitskreis.
 \end{itemize}
@@ -265,7 +268,7 @@ ZRM-EB:
 %}
 
 
-\textbf{Beispiel.} Bei beiden Methoden (EF, EB) erhält man das diskrete System durch entsprechendes Ersetzen von $s$ in $G(s)$, wie hier mit Hilfe der Symbolic Toolbox \texttrademark~ in Matlab\texttrademark~ durchgeführt. Die statische Verstärkung bleibt erhalten.
+\textbf{Beispiel.} Bei beiden Methoden (EF, EB) erhält man das diskrete System durch entsprechendes Ersetzen von $s$ in $G(s)$, wie hier mit Hilfe der Symbolic Toolbox \texttrademark~ in MATLAB\texttrademark~ durchgeführt. Die statische Verstärkung bleibt erhalten.
 
 \href{https://gitlab.cvh-server.de/jfalkenhain/diskretisierung-linearer-systeme/-/blob/master/Matlab/Diskretisierung_Euler.m}{File on Git: Diskretisierung\_Euler.m}
 
diff --git a/4_ZOH-Diskretisierung.tex b/4_ZOH-Diskretisierung.tex
index 1c8f62f..a252506 100644
--- a/4_ZOH-Diskretisierung.tex
+++ b/4_ZOH-Diskretisierung.tex
@@ -31,7 +31,7 @@ aber wegen der Sprungfunktion, also $u(k)=1$ für $k\geq 0$
 \[
 y(k+1)= y(k) + T u(k)~~,
 \] 
-womit die Differenzengleichung des Integrators ermittelt wäre. Diese \textit{verzögerte} Integration entspricht der des Euler-Forward-Verfahren mit der Übertragungsfuktion
+womit die Differenzengleichung des Integrators ermittelt wäre. Diese \textit{verzögerte} Integration entspricht der des Euler-Forward-Verfahren mit der Übertragungsfunktion
 \[
 G_I(z) =\frac{Y(z)}{U(z)} =\frac{T}{z-1}~~.
 \]
@@ -88,11 +88,11 @@ 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} \}~~.
+G_{d,ZOH}(z)= \frac{z-1}{z}\cdot Z\{ L^{-1}\left\{ \frac{1}{s} \cdot G(s)\right\}|_{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
+Tabelle \ref{table:impuls} liefert uns
 \[H(z)=\frac{Tz}{(z-1)^2}
 \]
 woraus wir
@@ -144,16 +144,16 @@ y^*(kT) &= \sum_{j=0}^{\infty} u(k) \underbrace{(h_k([k-j]T)-h_k([k-j-1]T))}_{
 %&+ \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 \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:
+Verwendet wird neben der Bedingung \eqref{ZOH:bed} die Faltungssumme 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{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.
+Durch den Abtaster entsteht 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) \} ~~.
+G_{d,ZOH}\{ G(s) \} = G_{d,Impuls}\left\{ \frac{1-e^{-sT}}{s} \cdot G(s) \right\} ~~.
 \end{align*}
 Eine aufwendigere Notation erklärt sich schrittweise von Innen nach Außen
 \begin{enumerate}
@@ -163,7 +163,7 @@ Eine aufwendigere Notation erklärt sich schrittweise von Innen nach Außen
 \item Z-Transformation der Folge ergibt die Übertragungsfunktion $G_{d,ZOH}(z)$
 \end{enumerate}
 \begin{align*}
-G_{d,ZOH}(z)= Z\{ L^{-1}\{ \frac{1-e^{-sT}}{s} \cdot G(s)\}|_{t=kT} \}~~.
+G_{d,ZOH}(z)= Z\{ L^{-1}\left\{ \frac{1-e^{-sT}}{s} \cdot G(s)\right\}|_{t=kT} \}~~.
 \end{align*}
 
 \textit{\textbf{Beispiel: Für den Integrator erhalten wir
@@ -223,15 +223,15 @@ ZOH-Diskretisiertes ZRM:
 \end{empheq}
 %}
 
-Darin stellt $\e^{\jv A T}$ eine Matrix-Exponetialfunktion dar: 
+%Darin stellt $\e^{\jv A T}$ eine Matrix-Exponentialfunktion dar: 
+%\begin{align*}
+%e^{\jv A_k T} = \sum_{i=0}^{\infty} \frac{\jv A^i T^i}{i!} %\text{MATLAB\texttrademark: \verb! expm(A)!}
+%\end{align*}
+Die numerische Berechnung der Matrix-Exponentialfunktion $e^{\jv A_k T}$, siehe Definition in \eqref{eq:MEF}, ist die Hauptaufgabe bei der Ermittlung des diskreten Modells (MATLAB\texttrademark: \verb! expm(Ak*T)!). Eine inverse Transformation
 \begin{align*}
-e^{\jv A_k T} = \sum_{i=0}^{\infty} \frac{\jv A^i T^i}{i!} %\text{MATLAB\texttrademark: \verb! expm(A)!}
+ \jv A_d =e^{\jv A_k T}  ~~\Leftrightarrow~~  \jv A_k = \frac{1}{T} \ln \jv A_d
 \end{align*}
-Ihre numerische Berechnung ist die Hauptaufgabe bei der Ermittlung des diskreten Modells (MATLAB\texttrademark: \verb! expm(A)!). 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 Matrix-Logarithmus nicht im Sinne einer elementweisen Operation, sondern als Matrixoperation zu verstehen ist (MATLAB\texttrademark: \verb! logm(A)!).
+ist jederzeit möglich, wobei auch der Matrix-Logarithmus nicht im Sinne einer elementweisen Operation, sondern als Matrixoperation zu verstehen ist (MATLAB\texttrademark: \verb! logm(Ad)!).
 
 
 %\begin{minipage}[b]{7cm}
diff --git a/5_Impulse-Diskretisierung.tex b/5_Impulse-Diskretisierung.tex
index b1a0b09..4566079 100644
--- a/5_Impulse-Diskretisierung.tex
+++ b/5_Impulse-Diskretisierung.tex
@@ -1,17 +1,19 @@
-\subsection{Impulse-Diskretisierung
+\subsection{Impuls-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
 \begin{align}\label{Impulse:bed}
-g_k(kT) =g_d (k)~~\laplace ~~G_{Impulse}(z)~~.
+g_k(kT) =g_d (k)~~\laplace ~~G_{Impuls}(z)~~.
 \end{align}
-In der Praxis lassen sich damit natürlich auch Linearisierungen, Systemvereinfachungen, Arbeitspunktmodelle usw. komplexerer Systeme ermitteln. Die Ordnung des diskreten Systems ergibt sich prinzipiell der aus Anzahl der verwendeten Abtastzeitpunkte. Die Übertragungsfunktion entsteht der aus z-Transformation der abgetasteten Impulsfolge.
+Als Entwurfsmethode für digitale Filter resultiert die Bezeichnung \textit{Impuls-invariante Methode}.
+
+In der Praxis lassen sich damit natürlich auch Linearisierungen, Systemvereinfachungen, Arbeitspunktmodelle usw. komplexerer Systeme ermitteln. Die Ordnung des diskreten Systems ergibt sich prinzipiell aus der Anzahl der verwendeten Abtastzeitpunkte. Die Übertragungsfunktion entsteht aus der z-Transformation der abgetasteten Impulsfolge.
 
 In 
 \cite{bild_quad} 
 %\url{https://www.eit.hs-karlsruhe.de/mesysto/fileadmin/downloads/teilB/PDFs/Musterloesungen/Skript_TDS_Musterloesungen_Kapitel_6.pdf}
 %\url{https://www.eit.hs-karlsruhe.de/mesysto/quicklink/startseite.html}
-wird in der konkreten Beispielaufgabe darauf hingewiesen, dass die statische Verstärkung $K_S=G_{Impulse}(1)$ nicht der des kontinuierlichen Systems entspricht, mit geringerer Abtastzeit aber immer besser approximiert wird. Eine Erklärung dafür wird auch im Frequenzbereich gesucht.
+wird in der konkreten Beispielaufgabe darauf hingewiesen, dass die statische Verstärkung $K_S=G_{Impuls}(1)$ nicht der des kontinuierlichen Systems entspricht, mit geringerer Abtastzeit aber immer besser approximiert wird. Eine Erklärung dafür wird auch im Frequenzbereich gesucht.
 
 Die Diskretisierung eines Durchgriffs oder statischen Anteils auf diese Weise ist wenig sinnvoll. Die Impulsantwort eines kontinuierlichen Systems erhält durch diesen den Anteil $g(t)=d\cdot \rho(0)+\tilde g(t)$. Bis auf diesen Impuls bei $t=0$ ist sie identisch mit $\tilde g(t)$, der Impulsantwort mit verschwindendem Durchgriff. Der (pseudo-) kontinuierliche Dirac-Impuls lässt sich wie eingangs erwähnt nicht sinnvoll abtasten. Bei Impuls-Diskretisierung könnte man den statischen Anteil einfach vom kontinuierlichen System übernehmen ($d_d=d_k$).
 
@@ -19,7 +21,7 @@ Die theoretische Ermittlung korrespondierender diskreter Systeme aus der kontinu
 
 
 \begin{table}[H]
-  \caption{Impuls-diskretisierte Systeme%
+  \caption{Laplace-und  Z-transformierte Signalpaare%
            \label{table:impuls}}
   \medskip
   \begin{center}
@@ -37,7 +39,7 @@ Die theoretische Ermittlung korrespondierender diskreter Systeme aus der kontinu
   \end{center}
 \end{table}
 
-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 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 Impuls-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. Einige Beispiele sind hier in MATLAB\texttrademark~ gezeigt:
 
@@ -67,8 +69,48 @@ woraus sich erkennen lässt, wenn man auf Pol-/Nullstellenform aufmultiplizieren
 \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 ein Nennerpolynom vom Grad $n$ entstehen würde und damit auch $n$ Nullstellen,
 \item bei $d= 0$ eine der Nullstellen $z_{i0}=0$ heißt.
 \end{enumerate}
 
+\textbf{Zustandsraummodell.} Mit den Parametern des Zustandsraummodells ausgedrückt, erhält man aus der allgemeinen Lösung \eqref{ZRM_kontLsg} für die Impulsantwort
+\begin{align*}%\label{ZRM_kontLsg}
+\jv g(t) = \jv C_k e^{\jv A_kt}\jv B_k  + \jv D_k \jv \rho(t)
+\end{align*}
+Der Dirac-Impuls lässt sich nicht abtasten, es kann aber gefordert werden, dass die diskrete Antwort einen entsprechenden diskreten Impuls zum Zeitpunkt $t=0$ enthält, also
+\begin{align*}%\label{ZRM_kontLsg}
+\jv g(kT) = \jv C_k e^{\jv A_k k T}\jv B_k  + \jv D_k \jv \rho(kT) = \left\{
+\begin{array}{l}
+\jv C_k \jv B_k +\jv D_k ~~\text{für}~k = 0\\
+\jv C_k e^{\jv A_k kT}\jv B_k ~~\text{für}~k > 0
+\end{array}\right.
+\end{align*}
+Die diskrete Impulsantwort erhält man aus der allgemeinen Lösung \eqref{ZRM_diskLsg} mit
+\begin{align*}%\label{ZRM_kontLsg}
+\jv g(k) = \jv C_d {\jv A_d ^{k-1} }\jv B_d  + \jv D_d \jv \rho(k) = \left\{
+\begin{array}{l}
+\jv D_d ~~\text{für}~k = 0\\
+\jv C_d \jv A_d^{k-1}\jv B_d ~~\text{für}~k > 0
+\end{array}\right.~.
+\end{align*}
+Die Gleichheit diskreter und kontinuierlicher Impulsantwort ist offensichtlich für $k=2$ für die folgende Wahl
+\begin{align*}%\label{ZRM_kontLsg}
+\jv C_k e^{2\jv A_k T}\jv B_k \stackrel{!}{=}  \underbrace{\jv C_d}_{\jv C_k} \underbrace{\jv A_d}_{e^{\jv A_k T}} \underbrace{A_d\jv B_d}_{\jv B_k}
+\end{align*}
+gegeben, was sich für $k>2$ bestätigt.
+Damit lautet die diskrete Systemmatrix $\jv A_d= e^{\jv A_k T}$  (mit der Matrix-Exponentialfunktion, siehe Gl. \eqref{eq:MEF}) und $\jv B_d = \jv A_d^{-1} \jv B_k$.
+Der Durchgriff des diskreten Systems erzeugt eine diskreten Impuls entsprechend der Amplitude des kontinuierlichen Systems, wobei fraglich bleibt wie sinnvoll das ist, und gleicht für $\jv D_k=\jv 0$ den diskreten Ausgang an den initialen Sprung des kontinuierlichen Systems bei $t=0$ an. Daraus ergeben sich aus dem Vergleich für $k=0$
+die zwei Anteile
+\begin{align*}%\label{ZRM_kontLsg}
+\jv D_d = \jv C_k \jv B_k +\jv D_k~~.
+\end{align*}
 
+Impuls-diskretisiertes ZRM:
+\begin{empheq}[box=\fbox]{align*}
+%\boxed{
+ \jv A_d &=e^{\jv A_k T}         ~~&~~ \jv B_d &= e^{-\jv A_k T} \jv B_k \\
+%&&= (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 \jv B_k +\jv D_k
+%}
+\end{empheq}
+%}
diff --git a/6_FOH-Diskretisierung.tex b/6_FOH-Diskretisierung.tex
index 9d39b8e..7e93329 100644
--- a/6_FOH-Diskretisierung.tex
+++ b/6_FOH-Diskretisierung.tex
@@ -36,7 +36,7 @@ r_k(kT) =r_d (k)~~\laplace ~~R(z)~~
 \begin{align*}%\label{Ramp:bed}
 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
+basierend auf der Linearität des Systems, da die Sprungfunktion auch eine Summe von Impulsen darstellt. Daraus lässt sich
 \begin{align*}%\label{Ramp:bed}
 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*}
@@ -102,8 +102,8 @@ Für ein System $G(s)$ lautet die Laplace-Transformierte der Rampenantwort
 \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.
-Eine Entsprechung im z-Bereich $R(z)$ kann eventuell einer Korrespondeztabelle entnommen bzw. mit deren Hilfe ermittelt werden. Über den Zusammenhang 
+da sie aus der zweifachen Integration des Dirac-Impulses entsteht.
+Eine Entsprechung im z-Bereich $R(z)$ kann eventuell einer Korrespondneztabelle entnommen bzw. mit deren Hilfe ermittelt werden. Über den Zusammenhang 
 \[
 R(z) = \frac{T}{z-1}H(z)
 \]
@@ -152,7 +152,7 @@ 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:
+\textbf{Zustandsraummodell.} 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*}
@@ -194,74 +194,3 @@ Hiermit können die angegebenen Matrizen gewonnen werden (wie im MATLAB\texttrad
 
 
 
-%\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}
-%\end{eqnarray}
-
-%\subsection{Compilieren%
-%            \label{sec:Compilieren}}
-%
-%\begin{figure}[H]
-%  \begin{center}
-%    \includegraphics[width=0.3\textwidth]{bilder/roll_pitch_yaw}
-%      % Einbinden einer Pixelgrafik.
-%      % Die Endung „.png“ darf weggelassen werden.
-%    \caption{Ein Beispielbild mit Quellenangabe \citep{yawpitchroll2013}%
-%             \label{fig:roll_pitch_yaw}}
-%  \end{center}
-%\end{figure}
-%
-%\begin{minipage}[b]{7cm}
-%  \centering
-%    \includegraphics[width=7cm]{bilder/quadrocopter}
-%    \captionof{figure}{Quadrocopter \newline
-%                         % kein \\ innerhalb von \caption oder \captionof
-%                         % \newline
-%                       \citep{bild_quad}%
-%	               \label{fig:quadrocopter}}
-%\end{minipage}\hfill
-%\begin{minipage}[b]{7cm}
-%  \centering
-%  \includegraphics[width=7cm]{bilder/ka32S}
-%  \captionof{figure}{Koaxialhelicopter \newline
-%                     \citep[S. 101]{hubschrauber1997}%
-%                     \label{fig:ka32S}}
-%\end{minipage}
-	
-
-%\begin{table}[H]
-%  \caption{Masse des anzuhebenden Trägers%
-%           \label{table:massen}}
-%  \medskip
-%  \begin{center}
-%    \begin{tabular}{l|r}
-%      %
-%      % Die Leerzeichen im Quelltext haben keinen Einfluß auf die
-%      % Anordnung innerhalb der Tabelle. Diese wird durch die o.a.
-%      % Angabe „{l|r}“ gesteuert: linksbündiges Feld, senkrechter
-%      % Strich, rechtsbündiges Feld.
-%      %
-%      Bauteil                  & Masse[g] \\
-%      \hline
-%      Trägerrohr               &       35 \\
-%      Linearlager              &        7 \\
-%      Lagerblock Linearlager   &        5 \\
-%      Kabel und Schrauben      &       20 \\
-%      Motoren                  &       50 \\
-%      Propeller                &       10 \\
-%      Propeller Eingriffschutz &       80 \\
-%      Holzplatte               &       14 \\
-%      Drehzahlsensoren         &        5 \\
-%      \hline
-%      Gesamtmasse              &      226 \\
-%      % @@@ PG: Stimmt nicht: Die Summe der Zahlen ist 194!
-%    \end{tabular}
-%  \end{center}
-%\end{table}
-%
-   
-%\glqq{}{Boris ermöglicht die blockorientierte Simulation
- %        dynamischer Systeme nahezu beliebiger Art}\grqq{}
- %        \citep[S. 25]{boris2009}.
-         
diff --git a/7_Tustin-Diskretisierung.tex b/7_Tustin-Diskretisierung.tex
index d664a06..af6b38d 100644
--- a/7_Tustin-Diskretisierung.tex
+++ b/7_Tustin-Diskretisierung.tex
@@ -74,12 +74,17 @@ Setzt man jedoch den Ausdruck für $s$ so in eine Übertragungsfunktion ein, erh
 s &= \frac{2}{T}\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*}
-Durch Abbruch nach dem ersten Summanden entsteht der bekannte Zusammenhang \eqref{eq:sTustin}.
+Durch Abbruch nach dem ersten Summanden entsteht der bekannte Zusammenhang \eqref{eq:sTustin}. 
 
-Unter dem Begriff der Padé-Approximation wird eine Reihenentwicklung in gebrochen rationalen Funktionen durchgeführt. Dies kann auch auf die e-Funktion angewendet werden und man erhält durch Beschränkung auf eine Approximation erster Ordnung den Zusammenhang \eqref{eq:sTustin} ebenfalls.
+Unter dem Begriff der Padé-Approximation wird eine Reihenentwicklung in gebrochen rationalen Funktionen durchgeführt. Dies kann auch auf die e-Funktion angewendet werden und man erhält durch Beschränkung auf eine Approximation erster Ordnung den Zusammenhang \eqref{eq:sTustin} ebenfalls. Dies müsste der Umformung
+\begin{align*}
+e^{sT} =e^{sT/2}\cdot e^{sT/2} =\frac{e^{sT/2}}{e^{-sT/2}} = \frac{\sum_{n=0}^\infty \frac{(T/2)^n}{n!} s^n}{\sum_{n=0}^\infty \frac{(-T/2)^n}{n!} s^n} \approx
+\frac{1+sT/2}{1-sT/2}
+\end{align*}
+entsprechen, wobei dei Exponentialfunktionen in Zähler und Nenner entsprechend \eqref{eq:TaylorE} entwickelt werden, siehe \cite{BrianDouglas}.
 
 
-\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
+\textbf{Differenziation.} Die beiden Euler-Methoden aus Kap. \ref{sec:Euler-Diskretisierung} wurden dadurch motiviert, dass der Differenzenquotient 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} 
 \end{align*}
@@ -108,6 +113,13 @@ G(z)&=K\frac{\Pi_{i=1}^q (\frac{2}{T}\frac{z-1}{z+1}-s_{i0})}{\Pi_{i=1}^n (\frac
 Auch hier rechnet man nach, dass
 \[G_k(0) K\frac{\Pi_{i=1}^q (-s_{i0})}{\Pi_{i=1}^n (-s_i)} =G_d(1) =K_s
 \]
+In \cite{BrianDouglas} wird dargestellt, dass diese bilineare Transformation die linke s-Halbebene genauso in den Einheitskreis der z-Ebene abbildet, wie $z=e^{sT}$ aus Bild \ref{fig:s-z}. Im Detail betrachtet, sind die Abbildungen aber natürlich verschieden, z. B.:
+\begin{align}
+&\lim_{s\rightarrow -\infty}e^{sT} = 0\\
+&\lim_{s\rightarrow -\infty}\frac{1+sT/2}{1-sT/2} = -1
+\end{align}
+Die Bildung neuer Nullstellen durch Diskretisierung kann, ähnlich wie bei Wurzelortskurven, durch die Abbildung von Nullstellen aus dem Unendlichen interpretiert werden.
+
 
 \textbf{Zustandsraummodell.} In \cite{WolframVogt} wird das ZRM im z-Bereich aus der kontinuierlichen Zustandsgleichung im Frequenzbereich
 \begin{align*}
@@ -214,8 +226,5 @@ wird aber gezeigt, dass die Systeme identisch sind.
 %  \end{center}
 %\end{table}
 %
-   
-%\glqq{}{Boris ermöglicht die blockorientierte Simulation
- %        dynamischer Systeme nahezu beliebiger Art}\grqq{}
- %        \citep[S. 25]{boris2009}.
+
          
diff --git a/8_Matched-Filter.tex b/8_Matched-Filter.tex
index f88f9ad..9151aa2 100644
--- a/8_Matched-Filter.tex
+++ b/8_Matched-Filter.tex
@@ -5,9 +5,9 @@ Ein Ansatz der Approximation ist es, Pole und Nullstellen entsprechend der Trans
 \begin{align}
 z_i = e^{s_i T}
 \end{align}
-für das diskrete System festzulegen. 
+für das diskrete System festzulegen.  
 
-Zur Anwendung in der Regelungstechnik wird der Ansatz  in \cite{unbehauen2000RT2}, S.128, kurz erwähnt. In \cite{Lunze16b}, S.511, wird unter der Bezeichnung "`Approximation des p/n Bildes"' genau das vorgeschlagen, wobei auch erwähnt wird, dass der verbleibende Parameter $K_s$ identisch zum kontinuierlichen System gewählt werden solle. Außerdem wird empfohlen, nur mit einem zeitdiskreten Regler $G_{Impuls}(z)$ mit (vermutlich maximalem) Polüberschuss von eins zu arbeiten. Sollte das kontinuierliche System weniger als $n-1$ Nullstellen haben, so möge man $n-q-1$ Nullstellen $z_{i0}=-1$ ergänzen.
+Zur Anwendung in der Regelungstechnik wird der Ansatz  in \cite{unbehauen2000RT2}, S.128, kurz erwähnt. In \cite{Lunze16b}, S.511, wird unter der Bezeichnung "`Approximation des p/n Bildes"' genau das vorgeschlagen, wobei auch erwähnt wird, dass der verbleibende Parameter $K_s$ identisch zum kontinuierlichen System gewählt werden solle. Außerdem wird empfohlen, nur mit einem zeitdiskreten Regler $G_{Impuls}(z)$ mit (vermutlich maximalem) Polüberschuss von eins zu arbeiten. Sollte das kontinuierliche System weniger als $n-1$ Nullstellen haben, so möge man $n-q-1$ Nullstellen $z_{i0}=-1$ ergänzen. Dies entspricht der Interpretation, dass sich bestimmte Nullstellen des kontinuierlichen Systems im (negativ) unendlichen befinden.
 
 Mit \verb!c2d(sys,T,'matched')! wird genau das mit MATLAB\texttrademark~realisiert.
 
diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf
index 0948135221e1cd1e0303a403aa4cb1b835902d8c..92a8582f2d529d3cf784821f42b82e8544c97204 100644
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diff --git a/Regelungstechnik.tex b/Regelungstechnik.tex
index a2dce4a..00273f3 100644
--- a/Regelungstechnik.tex
+++ b/Regelungstechnik.tex
@@ -2,7 +2,8 @@
 \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
+\textbf{Kontinuierlicher Regler.} Regler ermitteln in der Regel die Stellgröße, weshalb $u(t)$ jetzt Ausgangssignal des Reglers ist, in Abhängigkeit vom Regelfehler \[e(t)=w(t)-y_m(t)~~.\] 
+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}
@@ -11,11 +12,11 @@ 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})~~,
+G_{PI-DT_1}(s) = \frac{U(s)}{E(s)} = k_p\left(1+\frac{1}{T_Is}+\frac{T_Ds}{1+T_Vs}\right)~~,
 \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)~~,
+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}.
 
@@ -35,11 +36,11 @@ so dass hier nur die Messgröße über den D-Anteil geführt wird, siehe Bild \r
 \end{align} 
 Die verzögerte  Integration des Typ I entspricht der Integration bei Forward-Euler-Diskretisierung oder ZOH-Diskretisierung. Ob diese hier geeignet ist, ist fraglich und der Fokus soll 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}\\
+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\\
+u_k^P = k_P e_k
 \end{align} 
 womit sich der angegebene Regler zu
 \begin{align}
@@ -92,7 +93,7 @@ Für $T_V\rightarrow 0$ geht der Regler in die PID-Variante Gl. \eqref{eq:PIDLun
 
 Als Gedankenexperiment sei ein kontinuierlicher Regler ZOH-diskretisiert. Betrachtet man die Äquivalenz aus Bild \ref{fig:zeitbereichsäquivalenzZOH} liefert $G_{ZOH}(z)$ Werte, welche auf dem kontinuierlichen Ausgang der Reihenschaltung von ZOH-Glied und $G(s)$ liegt. Bei Einsatz von $G(z)$ wird dessen Ausgangsfolge wieder für die Abtastperiode konstant gehalten, woraus sich die Äquivalenz aus Bild \ref{fig:AllgemReglerdigital} ergibt. Das Ziel der Überlegung ist zu verdeutlichen, dass der Einbau von zwei Verzögerungsgliedern eine eher schlechte Lösung darstellt. Für den Grenzfall des P-Reglers ist aufgrund der separaten Abtastung ein Abtast-Halteglied unwirksam (siehe Reihenschaltung von MATLAB\texttrademark-ZOH-Blöcken), womit kein Widerspruch zu \ref{fig:PReglerdigital} vorliegt.
 
-\textbf{Methoden für die Regler-Diskretisierung.} Eine FOH-Diskretisierung ist offensichtlich die bessere Lösung, da die Werte von $G_{FOH}(z)$ auf dem kontinuierlichen Ausgang von $G(s)$ liegen, wenn dieses System mit einer linearen Interpolation zwischen den Abtastwerten angeregt wird. Diese Äquivalenz ist jedoch nicht realisierbar, da das FOH-Glied zur Interpolation nicht kausal ist. Der FOH-diskretisierte Regler ansich ist natürlich realisierbar. Tatsächlich wird Tustin-Approximation und  Matched-Filter-Approximation in \cite{Lunze16b} und \cite{unbehauen2000RT2} zur Reglerdiskretisierung vorgeschlagen, während Euler-Backward-Approximation nur der Vollständigkeit halber erwähnt wird und für den Differenzenquotienten und dessen PT1-Approximation (mit sehr kleiner Zeitkonstante) eingesetzt wird. FOH-Diskretisierung sollte in der Regel aber ebenso gut geeignet sein. 
+\textbf{Methoden für die Regler-Diskretisierung.} Eine FOH-Diskretisierung ist offensichtlich die bessere Lösung, da die Werte von $G_{FOH}(z)$ auf dem kontinuierlichen Ausgang von $G(s)$ liegen, wenn dieses System mit einer linearen Interpolation zwischen den Abtastwerten angeregt wird. Diese Äquivalenz ist jedoch nicht realisierbar, da das FOH-Glied zur Interpolation nicht kausal ist. Der FOH-diskretisierte Regler an sich ist natürlich realisierbar. Tatsächlich wird Tustin-Approximation und  Matched-Filter-Approximation in \cite{Lunze16b} und \cite{unbehauen2000RT2} zur Reglerdiskretisierung vorgeschlagen, während Euler-Backward-Approximation nur der Vollständigkeit halber erwähnt wird und für den Differenzenquotienten und dessen PT1-Approximation (mit sehr kleiner Zeitkonstante) eingesetzt wird. FOH-Diskretisierung sollte in der Regel aber ebenso gut geeignet sein. 
 
 \begin{figure}[H]
 \begin{center}
@@ -105,21 +106,21 @@ Als Gedankenexperiment sei ein kontinuierlicher Regler ZOH-diskretisiert. Betrac
 \textbf{Implementierung.} Liegt bereits eine Differenzengleichung oder ZRM vor, sollte die Implementierung keine Schwierigkeit darstellen. Aus der 
 z-Übertragungsfunktion ist eine Zeitbereichs-Darstellung zunächst abzuleiten. Die Darstellung als gebrochen rationale Funktion eignet sich, um direkt die Koeffizienten der kanonische Normalform eines digitalen Filters abzulesen. Stellt man die z-Übertragungsfunktion anstelle der Pol-/Nulstellenform \eqref{eq:diskPN} als gebrochen rationale Funktion
 \[
-G(z) = \frac{U(z)}{E(z)} =  \frac{\sum_{k=0}^n a_k z^{-k}}{\sum_{k=0}^q b_k z^{-k}}
+G(z) = \frac{U(z)}{E(z)} =  \frac{\sum_{k=0}^n b_k z^{-k}}{1 + \sum_{k=1}^q a_k z^{-k}}
 \]
 dar, dann ergibt sich im Zeitbereich
 \[
-b_0 e_k + b_1 e_{k-1} +b_2 e_{k-2} \hdots = a_0 u_k + a_1 u_{k-1} + a_2 u_{k-2} \hdots
+b_0 e_k + b_1 e_{k-1} +b_2 e_{k-2} \hdots = u_k + a_1 u_{k-1} + a_2 u_{k-2} \hdots
 \]
 Filter, realisiert durch die Filterstruktur Bild \ref{fig:Filter}. Die Filterkoeffizienten sind bis auf einen Faktor eindeutig.
 
-\textit{Beispiel: PI-DT1-Regler. Geht man von Gl. \eqref{eq:PIDT1} aus und setzt $b_0=1$, dann erhält man durch Aufmultiplizieren die auch in \cite{unbehauen2000RT2} angegebenen Filterkoeffizienten
-\begin{align}
-a_0 &= \frac{k_p}{1+T_V/T} \left(1+ \frac{T+T_V}{2T_I}  +  \frac{T_D+T_V}{T} \right)\\
-a_1 &= \frac{k_p}{1+T_V/T} \left(-1 + \frac{T}{2T_I} - 2\frac{T_D+T_V}{T} \right) \\
-a_2 &= \frac{k_p}{1+T_V/T} \left(\frac{T_D+T_V}{T}- \frac{T_V}{2T_I} \right)\\ 
-b_1 &= -\frac{T_V}{T+T_V} 
-\end{align}
+\textit{Beispiel: PI-DT1-Regler. Geht man von Gl. \eqref{eq:PIDT1} aus, dann erhält man durch Aufmultiplizieren die auch in \cite{unbehauen2000RT2} angegebenen Filterkoeffizienten
+\begin{align*}
+b_0 &= \frac{k_p}{1+T_V/T} \left(1+ \frac{T+T_V}{2T_I}  +  \frac{T_D+T_V}{T} \right)\\
+b_1 &= \frac{k_p}{1+T_V/T} \left(-1 + \frac{T}{2T_I} - 2\frac{T_D+T_V}{T} \right) \\
+b_2 &= \frac{k_p}{1+T_V/T} \left(\frac{T_D+T_V}{T}- \frac{T_V}{2T_I} \right)\\ 
+a_1 &= \frac{T}{T+T_V} ~~~a_2 = -\frac{T_V}{T+T_V} 
+\end{align*}
 Nachteilig gegenüber der parallelen Implementierung der drei Anteile P, I und DT1 ist, dass für den Grenzfall des verschwindenden I-Anteils ($T_I\rightarrow \infty$) die Differenzengleichung und die Filterstruktur formal weiterhin von zweiter Ordnung ist. Allerdings verhält das Filter sich exakt, wie das verbleibende Filter aus P- und DT1-Anteil erster Ordnung.     
 }
 				
diff --git a/anhang.tex b/anhang.tex
index a557dbb..a58983e 100644
--- a/anhang.tex
+++ b/anhang.tex
@@ -58,9 +58,9 @@ Benennt man jetzt den Zustand $\jv x_1 = \tilde{\jv x}$ um, kann man die Matrize
 \section{Spezielle Reihenentwicklung der Logarithmus-Funktion}
 \label{anh:Logarithmus}
 Die Taylor-Reihenentwicklung
-\begin{align*}%\label{ZRM_disk}
+\begin{align}\label{eq:Taylorreihe}
 f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}x_0}{n!} (x-x_0)^n
-\end{align*} 
+\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
@@ -75,7 +75,7 @@ Der Variablenwechsel soll dazu dienen, die Struktur in der folgenden Umformung w
 \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:
+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}
@@ -88,7 +88,7 @@ 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} \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)
+&= 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*}
 
  
diff --git a/hauptdatei.bib b/hauptdatei.bib
index d0b7d28..eb7f950 100644
--- a/hauptdatei.bib
+++ b/hauptdatei.bib
@@ -47,6 +47,14 @@ note = {\url{https://de.mathworks.com/help/control/ref/thiran.html#bsd5aup-2}}
 
 
 
+@MISC{BrianDouglas,
+author = {Brian Douglas},
+title = {{Discrete Control}},
+month = {März},
+year = {2018},
+note = {\url{https://www.youtube.com/watch?v=14cMhrp5wlk}}
+}
+
 @MISC{bild_quad,
 author = {Manfred Strohrmann},
 title = {{Systemtheorie Online}},
-- 
GitLab