From 3176ebfc2fe08dba0f8a41f179f9886ccd3ddf64 Mon Sep 17 00:00:00 2001
From: Jan Falkenhain <jan.falkenhain@hs-bochum.de>
Date: Thu, 13 May 2021 22:56:40 +0200
Subject: [PATCH] 15erd commit

---
 1_Einleitung.tex                    |  18 ++++---
 5_Impulse-Diskretisierung.tex       |  17 +++---
 7_Tustin-Diskretisierung.tex        |   2 +-
 FB_Diskretisierung.pdf              | Bin 2404788 -> 2401173 bytes
 Matlab/Diskretisierung_Impulse.m    |  77 +++++++++++++++++++---------
 Matlab/Diskretisierung_mit_MATLAB.m |  26 ++++++++--
 bilder/Impulsantwort_Vergleich.png  | Bin 34271 -> 34818 bytes
 bilder/Sprungantwort_Vergleich.png  | Bin 34552 -> 30252 bytes
 8 files changed, 95 insertions(+), 45 deletions(-)

diff --git a/1_Einleitung.tex b/1_Einleitung.tex
index 7f4c7c4..c516b14 100644
--- a/1_Einleitung.tex
+++ b/1_Einleitung.tex
@@ -4,21 +4,21 @@
 \textbf{Diskretisierung.} Diskrete Systeme werden im Rahmen der digitalen Signalverarbeitung und der Regelungstechnik entworfen, in letzterer zumeist mit Bezug zu kontinuierlichen, physikalischen Modellen. Zwei wesentliche Motive können hier vorliegen, um kontinuierliche Systeme zu diskretisieren, also sie durch ein mit fester Abtastzeit arbeitendes, diskretes System zu approximieren.
 
 \begin{enumerate}
-\item Hat man mit klassischen Methoden einen kontinuierlichen Regler entworfen, z. B. anhand eines Streckenmodells, dann erfordert die Realisierung auf einem Rechner (einer echtzeitfähigen Hardware beliebiger Art) eine diskrete Approximation, also ein unter Beachtung der Abtastzeit \textbf{den kontinuierlichen Regler} \textit{möglichst gut}  nachbildendes diskretes Modell. Dies ist immer erforderlich, wenn nicht Bibliotheken für die automatisierte Übersetzung kontinuierlicher Parameter, z. B. für PID-Regler, oder andere Software zur Verfügung steht, welche eine numerische Lösung bieten (wie z. B. MATLAB-Coder\texttrademark).  
+\item Hat man mit klassischen Methoden einen kontinuierlichen Regler entworfen, z. B. anhand eines Streckenmodells, dann erfordert die Realisierung auf einem Rechner (einer echtzeitfähigen Hardware beliebiger Art) eine diskrete Approximation, also ein unter Beachtung der Abtastzeit \textbf{den kontinuierlichen Regler} \textit{möglichst gut}  nachbildendes diskretes Modell. Dies ist immer erforderlich, wenn nicht Bibliotheken für die automatisierte Übersetzung kontinuierlicher Parameter, z. B. für PID-Regler, oder andere Software zur Verfügung stehen, welche eine numerische Lösung bieten (wie z. B. MATLAB-Coder\texttrademark).  
 \item Eine etwas andere Vorgehensweise des Reglerentwurfs besteht darin, den diskreten Regler direkt anhand eines diskreten Streckenmodells zu entwerfen. Dafür ist jetzt also die \textbf{Strecke bzw. das kontinuierliche Streckenmodell} zu diskretisieren. Anschließend steht eine eigene Welt von Methoden für den diskreten Reglerentwurf zur Verfügung, welche oft Ähnlichkeiten mit klassischen kontinuierlichen Methoden aufweisen, aber letztlich eigenständige Methoden darstellen.
 \end{enumerate}
 
-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,
+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 Reglers unterscheidet sich fundamental von der Aufgabe, den Regler \textit{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. Der gesamte Bereich der Diskretisierung in der Regelungstechnik wird in der Videoserie \cite{BrianDouglas} sehr geeignet dargestellt.
+welche auf "`schnellere"' diskrete Systeme führen, was auch für FOH-Diskretisierung (Kap. \ref{sec:FOH-Diskretisierung}) gelten müsste (so das Fazit des Autors). Der gesamte Bereich der Diskretisierung in der Regelungstechnik wird in der Videoserie \cite{BrianDouglas} sehr geeignet dargestellt.
 
-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.
+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 (Kap. \ref{sec:Impulse-Diskretisierung}) ab. 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. 
+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 am Rechner 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.
+Besonders einfach sind die Ansätze der Euler-Diskretisierung Kap. \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 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.  
+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 Darstellung der Differenzialgleichung, z. B. als Signalflussgraph, jeder Integrator durch den entsprechenden 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. Diese und weitere 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}
 
@@ -40,7 +40,9 @@ Die Ansätze können, für die hier behandelten lineare Systeme, sowohl im Zeit-
 											\begin{tabular}{@{}c@{}}$z_{i0} = \frac{2/T+s_{i0}}{2/T-s_{i0}}  ~\&~ (n-q)$\\Nullstellen bei $z_{i0}=-1$\end{tabular}
 											  & $=G_k(0)$\\
       \hline
-	Impuls         & Nein    &	- 		&$\displaystyle \frac{z}{z-1}$	 			&g(kT)=g(t)&$z_i=e^{s_i T}$ &  &$\displaystyle \lim_{T->0}G_d(1)=G_k(0)$\\
+	Impuls         & Nein    &	- 		&$\displaystyle \frac{z}{z-1}$	 			&g(kT)=g(t)&$z_i=e^{s_i T}$ &  & 
+	%$\displaystyle \lim_{T->0}G_d(1)=G_k(0)$
+	- \\
 			\hline
 	ZOH   										& Nein	&-  &		$\displaystyle T\frac{1}{z-1}$ 		&    h(kT)=h(t)&$z_i=e^{s_i T}$& &$=G_k(0)$\\
       \hline
diff --git a/5_Impulse-Diskretisierung.tex b/5_Impulse-Diskretisierung.tex
index 4566079..f490950 100644
--- a/5_Impulse-Diskretisierung.tex
+++ b/5_Impulse-Diskretisierung.tex
@@ -13,15 +13,20 @@ 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 $K_S=G_{Impuls}(1)$ 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.
+Allerdings nimmt die Diskrepanz mit auch mit größerer Auflösung, also kleinerem $T$ nicht ab.
 
-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 ($d_d=d_k$).
+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 ($d_{d,1}=d_k$). Gilt $d_k =0$, so benötigt man den diskreten Durchgriff ($d_{d2}$) aber, um den Sprung der kontinuierlichen Impulsantwort bei $t=0$ im Diskreten abzubilden:
+\[
+g_d(0) = d_d \stackrel{!}{=} c^T_k b_k = g_k(0)~~\text{bei} d_k=0
+\]
+Bei Ermittlung der ZRM-Parameter werden beide Anteile (entsprechen MATLAB\texttrademark-Implementierung) berücksichtigt, siehe später. 
 
 Die theoretische Ermittlung korrespondierender diskreter Systeme aus der kontinuierlichen Übertragungsfunktion kann zumeist direkt anhand von Korrespondenztabellen erfolgen:  
 
 
 \begin{table}[H]
-  \caption{Laplace-und  Z-transformierte Signalpaare%
+  \caption{Laplace- und  Z-transformierte Signalpaare%
            \label{table:impuls}}
   \medskip
   \begin{center}
@@ -59,11 +64,11 @@ g(t) = d \rho(t) + \sum_{i=0}^n b_i e^{s_i t} \sigma(t)
 \end{align}
 bzw. abgetastet, wenn man den Durchgriff übernimmt, um den diskreten Impuls zu gewichten
 \[
-g(kT) = d \rho(K) + \sum_{i=0}^n b_i e^{s_i kT} \sigma(kT)~~.
+g(kT) = d \rho(k) + \sum_{i=0}^n b_i e^{s_i kT} \sigma(kT)~~.
 \]
 Z-Transformation liefert
 \[
-G(z) = d  + \sum_{i=0}^n b_i \frac{z}{z-e^{s_i T}}
+G(z) = d  + \sum_{i=0}^n b_i \frac{z}{z-e^{s_i T}}~~,
 \]
 woraus sich erkennen lässt, wenn man auf Pol-/Nullstellenform aufmultiplizieren würde, dass
 \begin{enumerate}
@@ -75,7 +80,7 @@ woraus sich erkennen lässt, wenn man auf Pol-/Nullstellenform aufmultiplizieren
 
 \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)
+\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}
diff --git a/7_Tustin-Diskretisierung.tex b/7_Tustin-Diskretisierung.tex
index af6b38d..e387deb 100644
--- a/7_Tustin-Diskretisierung.tex
+++ b/7_Tustin-Diskretisierung.tex
@@ -81,7 +81,7 @@ Unter dem Begriff der Padé-Approximation wird eine Reihenentwicklung in gebroch
 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}.
+entsprechen, wobei dei Exponentialfunktionen in Zähler und Nenner entsprechend Gl. \eqref{eq:TaylorE} entwickelt werden, siehe \cite{BrianDouglas}.
 
 
 \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
diff --git a/FB_Diskretisierung.pdf b/FB_Diskretisierung.pdf
index 80502ada163b3ef159a9a6ffdacef6f7061b33c6..9cc554ed8bc7d6fae51efe0448aadd3db6954b86 100644
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diff --git a/Matlab/Diskretisierung_Impulse.m b/Matlab/Diskretisierung_Impulse.m
index 591b611..7643c1d 100644
--- a/Matlab/Diskretisierung_Impulse.m
+++ b/Matlab/Diskretisierung_Impulse.m
@@ -1,42 +1,69 @@
 % Diskretisierung linearer Systeme nach Impulse
 clear all; close all
 
-% Integrator
-sys_I = tf([1],[1,1])
+% kontinuierliches System
+if 1  % als Übetragungsfunktion
+    sys = tf([1 1],[1,1, 1])
+    
+    A=ss(sys).A; B=ss(sys).B; C=ss(sys).C; D=ss(sys).D
+else
+    % ZRM-Daten des kontinuierlichen Systems
+    %A=0.1*[1 -2; 3 -4];
+    %B=[1; 2];
+    %C=[2 1; 3 3];
+    %D=[2; 2];
 
+    %sys = ss(A,B,C,D);
+end
+
+%Abtastzeit
+T=0.1
+
+%% Symbolischer Weg über z-Transformation
 % aus System eine Funktion mit Variable s
 syms s z t
-[num, den]=tfdata(sys_I);
+[num, den]=tfdata(sys);
 fun_I = poly2sym(cell2mat(num),s) / poly2sym(cell2mat(den),s)
 
 % inverse Laplace-Transformation
 Tfun_I = ilaplace(fun_I)
-%Abtastzeit 
-T=0.1
 Tfun_I = subs(Tfun_I, t*T)
 
 %z_transformation
 Zfun_I = ztrans(Tfun_I)
 
 [num, den] = numden(Zfun_I)
-sys_d = T*tf(sym2poly(num),sym2poly(den),T)
-sys_d2 = c2d(sys_I, T, 'impulse')
-
-%Impulsantwort
-figure(1), impulse(sys_I), hold on
-[y,tt]=impulse(sys_d); plot(tt, y ,'*','color', 'magenta')
-[y,tt]=impulse(sys_d2); plot(tt, y ,'*','color', 'green')
-
-% Verstärkung
-dcgain(sys_d)
-dcgain(sys_d2)
-[num,den]=tfdata(sys_d2)
-zfun = poly2sym(cell2mat(num),z) / poly2sym(cell2mat(den),z)
-double(subs(zfun,1))
-
-
-% Sprungantwort
-figure(2), step(sys_I), hold on
-[y,tt]=step(sys_d); plot(tt, y ,'*','color', 'magenta')
-[y,tt]=step(sys_d); plot(tt, y ,'*','color', 'green')
+sys_d = tf(sym2poly(num),sym2poly(den),T)
+
+%% sys_d2M ist nach MATLAB-Impuls diskretisiertes System
+% Impulse, diskretisiert auf MATLAB Impuls 
+sys_d2M = c2d(sys, T, 'impulse')
+% 1/T* sys_dM ist ungefähr echter Impuls
+sys_d2 = sys_d2M *(1/T)
+
+%% Matrizen direkt berechnen
+A_d =expm(A*T)
+B_d = A_d*B 
+D_d = D + C*B
+C_d = C
+sys_d3 = ss(A_d,B_d,C_d,D_d,T)
+
+% sollte den Matrizen aus T*sys_d2 entsprechen (TRM nicht eindeutig)
+{ss(sys_d).A, ss(sys_d).B, ss(sys_d).C, ss(sys_d).D}
+
+%% Simulation des Impulses
+% MATLAB-Impuls rho_Matlab = 1/T*rho 
+figure(1)
+impulse(sys);hold on   %alle diskreten Systeme identisch
+[y,t]=impulse(T*sys_d);plot(t, y ,'*','color', 'blue')
+[y,t]=impulse(T*sys_d);plot(t, y ,'*','color', 'red')
+[y,t]=impulse(T*sys_d);plot(t, y ,'*','color', 'green')
+% 1/T *roh * T* sys_d..
+
+figure(2);impulse(sys);hold on
+% ALternative Impulssimulation, echten Impuls am Eingang
+u=0:T:100*T
+u=zeros(length(u),1); u(1)=1;
+
+[y,t]=lsim(sys_d,u);plot(t, y ,'*','color', 'blue')
 
diff --git a/Matlab/Diskretisierung_mit_MATLAB.m b/Matlab/Diskretisierung_mit_MATLAB.m
index ac3e30e..443ff25 100644
--- a/Matlab/Diskretisierung_mit_MATLAB.m
+++ b/Matlab/Diskretisierung_mit_MATLAB.m
@@ -8,7 +8,11 @@ sys = tf([1,1],[1 1 1])
 T=0.5;
 sys_ZOH = c2d(sys,T,'ZOH')  % default ohne 'methode'
 sys_FOH = c2d(sys,T,'FOH')  % default ohne 'methode'
-sys_impulse = c2d(sys,T,'impulse')  % default ohne 'methode'
+sys_impulseM = c2d(sys,T,'impulse')  % default ohne 'methode'
+
+% diskretisiert nach MATLAB-Impuls. Dieser hat für k=0 den Wert 1/T (statt 1). Damit
+sys_impulse = (1/T)*sys_impulseM
+
 sys_Tustin = c2d(sys,T,'Tustin')  % default ohne 'methode'
 
 % Vergleich anhand der Sprungantworten
@@ -20,6 +24,7 @@ figure(1), step(sys), hold on
 [y,t]=step(sys_Tustin); plot(t, y ,'*','color', 'green')
 
 legend({'','ZOH','FOH','Impulse','Tustin'},'Location','South')
+axis([0 10 0 3.2])
 % dcgain(sys_impulse) % statische Verstärkung nur für sys_impulse ungleich
 % 1 =dcgain(sys), deshalb die Abweichung 
 
@@ -29,11 +34,22 @@ legend({'','ZOH','FOH','Impulse','Tustin'},'Location','South')
 %% Jetzt Impuls-Antworten berechnen und vergleichen
 figure(2), impulse(sys), hold on
 
-[y,t]=impulse(sys_ZOH); plot(t, y ,'*','color', 'blue')
-[y,t]=impulse(sys_FOH); plot(t, y ,'*','color', 'red')
-[y,t]=impulse(sys_impulse); plot(t, y ,'*','color', 'black')
-[y,t]=impulse(sys_Tustin); plot(t, y ,'*','color', 'green')
+% impulse berechnet nach MATLAB für den Impuls 1/T. Impuls der Höhe 1,
+% deshalb für T*y
+[y,t]=impulse(sys_ZOH); plot(t, T*y ,'*','color', 'blue')
+[y,t]=impulse(sys_FOH); plot(t, T*y ,'*','color', 'red')
+[y,t]=impulse(sys_impulse); plot(t, T*y ,'*','color', 'black')
+[y,t]=impulse(sys_Tustin); plot(t, T*y ,'*','color', 'green')
 legend({'','ZOH','FOH','Impulse','Tustin'},'Location','North')
 
 % Bild speichern
 % saveas(gcf,'../bilder/Impulsantwort_Vergleich.png')
+
+% Werte der Impulsantwort für k=0, t=0, nur für das Impulssystem gilt
+% Gleichheit
+if(ss(sys_impulse).D == ss(sys).C * ss(sys).B)
+    disp('für k=0 wird der Wert des kont. Systems getroffen')
+end
+
+(ss(sys_Tustin).D == ss(sys).C * ss(sys).B)
+   
\ No newline at end of file
diff --git a/bilder/Impulsantwort_Vergleich.png b/bilder/Impulsantwort_Vergleich.png
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