From 92e8f57c184c6573e27d06bfbbc432b086777314 Mon Sep 17 00:00:00 2001
From: Jan Falkenhain <jan.falkenhain@hs-bochum.de>
Date: Thu, 22 Jul 2021 21:08:49 +0200
Subject: [PATCH] 19erd commit

---
 5_Impulse-Diskretisierung.tex |   4 +++-
 6_FOH-Diskretisierung.tex     |   8 +++----
 FB_Diskretisierung.pdf        | Bin 2419717 -> 2259390 bytes
 Regelungstechnik.tex          |  43 +++++++++++++++++++++++++++-------
 bilder/Filter.PNG             | Bin 217234 -> 22979 bytes
 5 files changed, 41 insertions(+), 14 deletions(-)

diff --git a/5_Impulse-Diskretisierung.tex b/5_Impulse-Diskretisierung.tex
index afd58a2..d3af5cb 100644
--- a/5_Impulse-Diskretisierung.tex
+++ b/5_Impulse-Diskretisierung.tex
@@ -36,7 +36,9 @@ Die theoretische Ermittlung korrespondierender diskreter Systeme aus der kontinu
       $\frac{1}{s}$               &  $\sigma(t)$     &  $\sigma(k)$ & $\frac{z}{z-1}$ \\
 			$\frac{1}{s^2}$               &  $t\cdot \sigma(t)$     &  $kT\cdot\sigma(k)$ & $\frac{Tz}{(z-1)^2}$ \\
 			$\frac{1}{s^3}$             &    $1/2 \cdot t^2$ &  $1/2\cdot(kT)^2$&    $\frac{T^2}{2}\frac{z(z+1)}{(z-1)^3}$ \\      
-      $\frac{1}{1+s \tau }$              &    $1/\tau  \cdot e^{-t/\tau}$ &  $1/\tau  \cdot e^{-kT/\tau}$&    $\frac{1}{\tau}\frac{z}{z-e^{-T/\tau}}$ \\      
+      $\frac{1}{1+s \tau }$              &    $1/\tau  \cdot e^{-t/\tau}$ &  $1/\tau  \cdot e^{-kT/\tau}$&    $\frac{1}{\tau}\frac{z}{z-e^{-T/\tau}}$ \\     
+			$\frac{a}{s(s+a)}$              &    $1-t\cdot e^{-at}$ &  $1-kT\cdot e^{-akT}$&    $\frac{(1-e^{-aT})z}{(z-1)(z-e^{-aT})}$ \\      
+
       \hline
       
       % @@@ PG: Stimmt nicht: Die Summe der Zahlen ist 194!
diff --git a/6_FOH-Diskretisierung.tex b/6_FOH-Diskretisierung.tex
index 335fc09..a37356b 100644
--- a/6_FOH-Diskretisierung.tex
+++ b/6_FOH-Diskretisierung.tex
@@ -108,9 +108,9 @@ Eine Entsprechung im z-Bereich $R(z)$ kann eventuell einer Korrespondneztabelle
 R(z) = \frac{T}{z-1}H(z)
 \]
 erhalten wir dann
-\[
+\begin{align}\label{eq:GFOHz}
 G_{FOH}(z) = \underbrace{\frac{z-1}{T}}_{\Laplace~ \gamma(k) \text{aus \eqref{eq:gammak}}}  \frac{z-1}{z} R(z)~~.
-\]
+\end{align}
 
 
 \textit{\textbf{Beispiel: Die Übertragungsfunktion des Integrators lautet $G(s)=1/s$. Wir suchen nach der $\mathscr{Z}$-Transformierten der Abtastung von $R(s)=1/s^3$. 
@@ -139,12 +139,12 @@ Eine andere Notation erklärt sich schrittweise von Innen nach Außen
 \item $\mathscr{Z}$-Transformation der Folge ergibt die Übertragungsfunktion $G_{d,FOH}(z)$
 \end{enumerate}
 \begin{align*}
-G_{d,ZOH}(z)= \mathscr{Z}\{ \mathscr{L}^{-1}\{ \frac{1}{Ts^2}  (e^{sT} + 2 + e^{-sT})\cdot G(s)\}|_{t=kT} \}~~.
+G_{d,ZOH}(z)= \mathscr{Z}\{ \mathscr{L}^{-1}\{ \frac{1}{Ts^2}  (e^{sT} - 2 + e^{-sT})\cdot G(s)\}|_{t=kT} \}~~.
 \end{align*}
 
 
 \textit{\textbf{Beispiel: Für den Integrator erhalten wir
-\[G_{FOH}(s)\cdot G(s) = \frac{1}{Ts^3}  (e^{sT} + 2 + e^{-sT})
+\[G_{FOH}(s)\cdot G(s) = \frac{1}{Ts^3}  (e^{sT} - 2 + e^{-sT})
 \]
 Mit Hilfe von Tabelle \ref{table:impuls} erhält man direkt die zugehörige z-Übertragungsfunktion für $1/s^3$ und damit
 \[
diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf
index 8a723553a0b2527fb8db432561194b8c16ac6e78..0f39e7586410442cd3750743d34d3ff501ffa581 100644
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diff --git a/Regelungstechnik.tex b/Regelungstechnik.tex
index d023d15..68e2fd6 100644
--- a/Regelungstechnik.tex
+++ b/Regelungstechnik.tex
@@ -97,12 +97,15 @@ In \cite{Isermann} wird beispielsweise zunächst der PID-Regler vom Typ I Gl. \e
 			\hline
 		Typ 1            & \cite{unbehauen2000RT2} 		&   $kp\left(1 + \frac{T}{2T_I} \frac{z+1}{z-1} +\frac{T_D}{T} \frac{z-1}{z(1+T_V/T)-T_V/T}  \right) $  		      \\
 		\hline
-    	Typ 2            & \cite{Isermann} 		&   $kp\left(1 + \frac{T}{T_I} \frac{1}{z-1} +\frac{T_D}{T} \frac{z-1}{z-e^{-T/T_V}}  \right) $    		          \\
+    	Typ 2            & \cite{Isermann} 		&   $kp\left(1 + \frac{T}{T_I} \frac{1}{z-1} +\frac{T_D}{T_V} \frac{z-1}{z-e^{-T/T_V}}  \right) $    		          \\
     \hline
-		Typ 3            & \cite{Isermann} 		&   $kp\left(1 + \frac{T}{T_I} \frac{z+1}{2(z-1)} +\frac{2 T_D(z-1)}{(T+2T_V)z + (T-2T_V)} \right) $    		          \\
+		Typ 3            & \cite{Isermann} 		&   $kp\left(1 + \frac{T}{2T_I} \frac{z+1}{(z-1)} +\frac{2 T_D(z-1)}{(T+2T_V)z + (T-2T_V)} \right) $    		          \\
     \hline
 		Typ 4            & \cite{documentationsimulation} 		&   $kp\left(1 +\frac{T}{T_I}\frac{1}{z-1} + \frac{T_D}{T} \frac{z-1}{z-1+{T/T_V}} \right) $    		          \\
     \hline
+		Typ 5            & siehe Bsp. FOH-diskretisiert 		&   $kp\left(1 + \frac{T}{2T_I} \frac{z+1}{(z-1)} +     \frac{T_D}{T} \frac{(z-1)(1-e^{-T/T_V})}{(z-e^{-T/T_V})} \right) $   		          \\
+    
+		\hline
 		
 		\end{tabular}
   \end{center}
@@ -156,15 +159,37 @@ 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} \hdo
 \]
 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, dann erhält man durch Aufmultiplizieren die auch in \cite{unbehauen2000RT2} angegebenen Filterkoeffizienten
+%\textit{
+Beispiel: PI-DT1-Regler als FOH-Diskretisierung 
+(Typ 5).
+Wie bereits gezeigt, führt die Diskretisierung des Integrators im FOH-Sinn auf das selbe System wie Tustin-Diskretisierung, weshalb
+\begin{align*}
+G_{PI} (z)= k_p \left(1+\frac{T}{2T_I}\frac{z+1}{z-1}\right)
+\end{align*}
+diekt angegeben werden kann. Aus dem Differentialanteil (DT1) $G_D(s)$ sei zunächst die Rampenantwort
+\begin{align*}
+R(s)=\frac{1}{s^2} G_D(s) = \frac{T_D}{s(1+T_Vs)} = T_D\frac{1/T_V}{s(1/T_V+s)}
+\end{align*}
+gebildet. Die letzte Darstellung eignet sich, um aus der letzten Zeile von Tabelle \ref{table:impuls} die $\mathscr{Z}$-Transformierte der diskreten Rampenantwort abzulesen:
+\begin{align*}
+R(z) = T_D \frac{(1-e^{-T/T_V})z}{(z-1)(z-e^{-T/T_V})}
+\end{align*}
+Nach Gl. \eqref{eq:GFOHz} folgt daraus
 \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} 
+G_{D-FOH}(z) = \frac{z-1}{T} \frac{z-1}{z} R(z) = \frac{T_D}{T} \frac{(z-1)(1-e^{-T/T_V})}{(z-e^{-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.     
-}
+
+ 
+%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.     
+
+%}
 				
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Abtastsystem, (ZOH-)Diskretisierung der Strecke 
diff --git a/bilder/Filter.PNG b/bilder/Filter.PNG
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