From 65388b40e0c8d8006e1111e9a100ee7250789a92 Mon Sep 17 00:00:00 2001
From: Jan Falkenhain <jan.falkenhain@hs-bochum.de>
Date: Tue, 11 May 2021 22:42:54 +0200
Subject: [PATCH] 14erd commit

---
 FB_Diskretisierung.pdf | Bin 2403261 -> 2404788 bytes
 Regelungstechnik.tex   |   8 +++++---
 hauptdatei.bib         |   8 ++++++++
 3 files changed, 13 insertions(+), 3 deletions(-)

diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf
index 92a8582f2d529d3cf784821f42b82e8544c97204..80502ada163b3ef159a9a6ffdacef6f7061b33c6 100644
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diff --git a/Regelungstechnik.tex b/Regelungstechnik.tex
index 00273f3..b9dd560 100644
--- a/Regelungstechnik.tex
+++ b/Regelungstechnik.tex
@@ -31,7 +31,7 @@ so dass hier nur die Messgröße über den D-Anteil geführt wird, siehe Bild \r
 
 \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 I:}~~ &G_I(z)=\frac{k_p}{T_I}\frac{T}{z-1}\\\label{eq:TypI}
 \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 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
@@ -42,7 +42,7 @@ gebildet (entsprechend Backward-Euler-Diskretisierung) und mit dem P-Anteil komp
 \begin{align}
 u_k^P = k_P e_k
 \end{align} 
-womit sich der angegebene Regler zu
+womit sich der angegebene Regler (Typ II) 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}}\\
@@ -52,7 +52,7 @@ Tabelliert erhält man in \cite{lutz:2005} für beide Typen die Koeffizienten de
 
 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 
 \begin{align}\label{eq:PIDLunze}
-U(z) = kp\left(1 + \frac{T}{2T_I} \frac{z+1}{z-1} +\frac{T_D}{T} \frac{z-1}{z}  \right)E(z)~~,
+\text{Typ III:}~~U(z) = kp\left(1 + \frac{T}{2T_I} \frac{z+1}{z-1} +\frac{T_D}{T} \frac{z-1}{z}  \right)E(z)~~,
 \end{align}
 wobei der Differenzialanteil nach
 \begin{align}\label{eq:BWE} 
@@ -68,6 +68,8 @@ G_{PI-DT1}(z) = kp\left(1 + \frac{T}{2T_I} \frac{z+1}{z-1} +\frac{T_D}{T} \frac{
 \end{align}
 Für $T_V\rightarrow 0$ geht der Regler in die PID-Variante Gl. \eqref{eq:PIDLunze} über.
 
+In \cite{Isermann} wird beispielsweise zunächst der PID-Regler vom Typ I \eqref{eq:TypI} angegeben mit der als Rechteckintegration bezeichneten verzögerten Integration (S. 108). Anschließend wird die Trapezintegration vorgeschlagen und Regler Typ III angegeben (S. 110). Unter dem Title "`PID-Regelalgorithmus durch Z-Transformation"' (S. 122) wird der ZOH-diskretisierte PI-DT1-Regler angegeben mit dem Hinweis auf "`gültige Parameter"' bei großen Abtastzeiten. Welche besondere Eignung der Regler haben könnte, auch wenn er bei einem vorgeschalteten ZOH-Glied vor die kontinuierliche Version dessen kontinuierlichen Ausgang exakt trifft, bleibt unklar. Im Folgenden (S. 126) wird der PI-DT1-Regler ensprechend Tustin-Diskretisierung approximiert und die Koeffizienten der rekursiven Differenzengleichung angegeben.   
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Diskretisierung und Implementierung allgemeiner Regler
          \label{sec:allgemeineRegler}}
diff --git a/hauptdatei.bib b/hauptdatei.bib
index eb7f950..02b26bc 100644
--- a/hauptdatei.bib
+++ b/hauptdatei.bib
@@ -27,6 +27,14 @@ note = {Lecture 7: Discrete Approximation of Continuous-Time Systems. \url{https
   year = 2006
 }
 
+@book{Isermann,
+  author = {Isermann, Rolf},
+  publisher = {Springer Vieweg},
+  series = {Lehrbuch},
+  title = {Digitale Regelsysteme, Band 1},
+  year = 1988
+}
+
 
 MISC{SystemtheorieOnline,
 author = {{Manfred Strohrmann}},
-- 
GitLab