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 \[ diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf index 8a723553a0b2527fb8db432561194b8c16ac6e78..0f39e7586410442cd3750743d34d3ff501ffa581 100644 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\frac{z-1}{z-e^{-T/T_V}} \right) $ \\ + Typ 2 & \cite{Isermann} & $kp\left(1 + \frac{T}{T_I} \frac{1}{z-1} +\frac{T_D}{T_V} \frac{z-1}{z-e^{-T/T_V}} \right) $ \\ \hline - Typ 3 & \cite{Isermann} & $kp\left(1 + \frac{T}{T_I} \frac{z+1}{2(z-1)} +\frac{2 T_D(z-1)}{(T+2T_V)z + (T-2T_V)} \right) $ \\ + Typ 3 & \cite{Isermann} & $kp\left(1 + \frac{T}{2T_I} \frac{z+1}{(z-1)} +\frac{2 T_D(z-1)}{(T+2T_V)z + (T-2T_V)} \right) $ \\ \hline Typ 4 & \cite{documentationsimulation} & $kp\left(1 +\frac{T}{T_I}\frac{1}{z-1} + \frac{T_D}{T} \frac{z-1}{z-1+{T/T_V}} \right) $ \\ \hline + Typ 5 & siehe Bsp. FOH-diskretisiert & $kp\left(1 + \frac{T}{2T_I} \frac{z+1}{(z-1)} + \frac{T_D}{T} \frac{(z-1)(1-e^{-T/T_V})}{(z-e^{-T/T_V})} \right) $ \\ + + \hline \end{tabular} \end{center} @@ -156,15 +159,37 @@ b_0 e_k + b_1 e_{k-1} +b_2 e_{k-2} \hdots = u_k + a_1 u_{k-1} + a_2 u_{k-2} \hdo \] 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 index 7045198d399552e88da9227c017c01e5c85161ac..0eb747f162468b9789819caa3a91304324158ffe 100644 GIT binary patch literal 22979 zcmeAS@N?(olHy`uVBq!ia0y~yV6tOiV7$b^#=yYvCg$J?1_lPk;vjb?hIQv;UNSH+ zu%tWsIx;Y9?C1WI$jZRLz**oCS<Jv7R06_`_tNAm7#J87OI#yLoQqNuOEUBG6hbm{ zQyGj5j0~*|%&d&e6$}lnj4iB;ELhn}uP`t~YJ0jkhE&A8z00}B=j;FDAN$|@UZ>+3 zsN!{Lokrb?EuNVRUAbGAczEp(%e|!J>H1UotG4czU!HZMxm_x%m*TE%+I`CCe`Wo9 z%j-Lyy}Lcfa{cqig>uHv&&0$$w|@S6-utpH*F`)G3<@mYKFKjMFies=$j-pf!bS(O z!D&m@Rjv1tHM7lfH%*x$lF$FM_W?75gW^o<@^yDB9{0ZcDH<M=sO;WXaz6R*zu)i6 z4zh|@RG-Lak!N7wI(aobK6m^5y4}_LN?(Wlem=kc&gY0;Y4dGaS6AJfq8YqL`GkL~ zP|cN;<;!-TNxAxkoxx#I^~<HxbMEb_yf^RW@Av!jFE95mKYzdSx$NqgotsQEE*yA! 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