From b885ada977c6e8419c459124cc56ae7f1ea3aded Mon Sep 17 00:00:00 2001
From: Jan Falkenhain <jan.falkenhain@hs-bochum.de>
Date: Mon, 26 Apr 2021 20:55:52 +0200
Subject: [PATCH] 12erd commit

---
 FB_Diskretisierung.pdf | Bin 2390225 -> 2390348 bytes
 Regelungstechnik.tex   |  10 +++++-----
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf
index e8ff27697183db566154c1444d13a7a85403469d..0948135221e1cd1e0303a403aa4cb1b835902d8c 100644
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diff --git a/Regelungstechnik.tex b/Regelungstechnik.tex
index 819b750..a2dce4a 100644
--- a/Regelungstechnik.tex
+++ b/Regelungstechnik.tex
@@ -33,7 +33,7 @@ so dass hier nur die Messgröße über den D-Anteil geführt wird, siehe Bild \r
 \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
+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}\\
 \end{align} 
@@ -55,12 +55,12 @@ U(z) = kp\left(1 + \frac{T}{2T_I} \frac{z+1}{z-1} +\frac{T_D}{T} \frac{z-1}{z}
 \end{align}
 wobei der Differenzialanteil nach
 \begin{align}\label{eq:BWE} 
-s\equiv  \frac{z-1}{zT}~~\text{Backward-Euler}
+s\equiv  \frac{z-1}{zT}~~\text{(Backward-Euler)}
 \end{align}
 diskretisiert bleibt.
 
 In  \cite{unbehauen2000RT2} wird der PID-T1-Regler angegeben. Dabei sei es auch hier von Vorteil, das PT1 Glied mit der Entsprechung
-nach Gl. \ref{Backward-Euler}
+nach Gl. \eqref{eq:BWE}
 zu diskretisieren:
 \begin{align}\label{eq:PIDT1}
 G_{PI-DT1}(z) = 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)
@@ -90,9 +90,9 @@ Für $T_V\rightarrow 0$ geht der Regler in die PID-Variante Gl. \eqref{eq:PIDLun
 \end{center}
 \end{figure}
 
-Als Gedankenexperiment sei ein kontinuierlicher Regler ZOH-diskretisiert. Betrachte 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 $G\textbf{}(z)$ 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 Verzögerungsglieder eine eher schlechte Lösung darstellt. Für den Grenzfall des P-Reglers ist aufgrund der separaten Abtastung ein Abtast-Haltglied unwirksam (siehe Reihenschaltung von MATLAB\texttrademark-ZOH-Blöcken), womit kein Widerspruch zu \ref{fig:PReglerdigital} vorliegt.
+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. 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 und für den Differenzenquotienten erwähnt 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 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. 
 
 \begin{figure}[H]
 \begin{center}
-- 
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