update notebook V02.1.ipynb

730e9c13 · Sebastian Schwarz · 774d9d2c · 730e9c13
Commit 730e9c13 authored 10 months ago by Sebastian Schwarz
--- a/sys2-jupyter-notebooks/exam_examples/V02.1.ipynb
+++ b/sys2-jupyter-notebooks/exam_examples/V02.1.ipynb
@@ -82,7 +82,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
@@ -156,16 +156,6 @@
      "     0     6    24    78   240\n",
      "\n"
     ]
-    },
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<IPython.core.display.Image object>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
    }
   ],
   "source": [
@@ -196,7 +186,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
@@ -235,7 +225,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
@@ -266,7 +256,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
@@ -295,24 +285,54 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
-      "Symbolic pkg v2.9.0: Python communication link active, SymPy v1.5.1.\n",
-      "F1 = (sym)\n",
+      "Symbolic pkg v2.8.0: Python communication link active, SymPy v1.5.1.\n"
+     ]
+    }
+   ],
+   "source": [
+    "pkg load symbolic\n",
+    "syms z"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "F = (sym)\n",
      "\n",
      "      6⋅z     \n",
      "  ────────────\n",
      "   2          \n",
      "  z  - 4⋅z + 3\n",
      "\n",
-      "F2 = (sym)\n",
+      "F1 = (sym)\n",
+      "\n",
+      "       6      \n",
+      "  ────────────\n",
+      "   2          \n",
+      "  z  - 4⋅z + 3\n",
+      "\n",
+      "G1 = (sym)\n",
      "\n",
-      "      3       9  \n",
+      "      3       3  \n",
+      "  - ───── + ─────\n",
+      "    z - 1   z - 3\n",
+      "\n",
+      "Gs = (sym)\n",
+      "\n",
+      "     3⋅z     3⋅z \n",
      "  - ───── + ─────\n",
      "    z - 1   z - 3\n",
      "\n"
@@ -320,11 +340,10 @@
    }
   ],
   "source": [
-    "pkg load symbolic\n",
-    "\n",
-    "syms z\n",
-    "F1 = 6*z/(z^2 - 4*z +3)\n",
-    "F2 = partfrac(F1)\n"
+    "F= 6*z/(z^2-4*z+3)\n",
+    "F1 = F*1/z\n",
+    "G1 = partfrac(F1)\n",
+    "Gs = expand(G1*z)"
   ]
  },
  {
@@ -372,13 +391,6 @@
    "plot(k,x2(1:5))"
   ]
  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  },
  {
   "cell_type": "code",
   "execution_count": null,

 %% Cell type:markdown id: tags:

 # <span style='color:OrangeRed'>V2 Z-TRANSFORMATION </span>

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">

 Hier vergleichen und überprüfen wir verschiedene Methoden zur Berechnung der Inverse der z-Transformation.

 %% Cell type:code id: tags:

 ``` octave
 % Necessary to use control toolbox
 pkg load control
 clear all
 ```

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">

 Ein bestimmtes Signal wird durch seine z-Transformation beschrieben, für die wir Zähler und Nenner kennen.

 %% Cell type:code id: tags:

 ``` octave
 num = [6 0]
 den = [1 -4 3]
 G = tf(num,den,1)
 ```

 %% Output

    num =
    
       6   0
    
    den =
    
       1  -4   3
    
    
    Transfer function 'G' from input 'u1' to output ...
    
               6 z
     y1:  -------------
          z^2 - 4 z + 3
    
    Sampling time: 1 s
    Discrete-time model.

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">

 Erste Option ist eine Potenzreihenentwicklung, d.h. wir wenden die Division von Zähler und Nenner mehrfach an:

 %% Cell type:code id: tags:

 ``` octave
 k=0:4;
 [x(1) r] = deconv(num,den)

 for j = 1:4
    numnew = conv(r,[1 0])
    [p r] = deconv(numnew,den)
    x(j+1) = p(length(p))
 end

 plot(k,x)


 ```

 %% Output

    x = 0
    r =
    
       6   0
    
    numnew =
    
       6   0   0
    
    p =  6
    r =
    
        0   24  -18
    
    x =
    
       0   6
    
    numnew =
    
        0   24  -18    0
    
    p =
    
        0   24
    
    r =
    
        0    0   78  -72
    
    x =
    
        0    6   24
    
    numnew =
    
        0    0   78  -72    0
    
    p =
    
        0    0   78
    
    r =
    
         0     0     0   240  -234
    
    x =
    
        0    6   24   78
    
    numnew =
    
         0     0     0   240  -234     0
    
    p =
    
         0     0     0   240
    
    r =
    
         0     0     0     0   726  -720
    
    x =
    
         0     6    24    78   240
    

-
-
 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">

 Zunächst beginnen wir mit der Partialbruchzerlegung. In Matlab/Octave gibt es die Funktion <code>residue</code>.
 Diese Funktion wird verwendet, um einen Bruch in Terme des Typs a/(z-b) zu zerlegen.
 Dann müssen wir zunächst den Zähler durch z dividieren und dann die Funktion verwenden.

 %% Cell type:code id: tags:

 ``` octave
 numz = deconv(num, [1 0])
 [r,p,q]=residue(numz,den)
 ```

 %% Output

    numz =  6
    r =
    
       3.0000
      -3.0000
    
    p =
    
       3
       1
    
    q = [](0x0)

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 r sind die Koeffizienten für die Zähler, p sind die Pole

 das Signal ist dann die Summe von Termen des Typs r*p^k

 %% Cell type:code id: tags:

 ``` octave
 npoles = length(p)

 for i=1:npoles
   s(i,:) = r(i).*p(i).^k
 end

 ```

 %% Output

    npoles =  2
    s =
    
         3.0000     9.0000    27.0000    81.0000   243.0000
    
    s =
    
         3.0000     9.0000    27.0000    81.0000   243.0000
        -3.0000    -3.0000    -3.0000    -3.0000    -3.0000
    

 %% Cell type:code id: tags:

 ``` octave
 plot(k,s(1,:),k,s(2,:),k,s(1,:)+s(2,:))
 ```

 %% Output



 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">

 Wir können das Ergebnis auch mit Hilfe der symbolischen Toolbox überprüfen.

 %% Cell type:code id: tags:

 ``` octave
 pkg load symbolic
-
 syms z
-F1 = 6*z/(z^2 - 4*z +3)
-F2 = partfrac(F1)
 ```

 %% Output

-    Symbolic pkg v2.9.0: Python communication link active, SymPy v1.5.1.
-    F1 = (sym)
+    Symbolic pkg v2.8.0: Python communication link active, SymPy v1.5.1.
+
+%% Cell type:code id: tags:
+
+``` octave
+F= 6*z/(z^2-4*z+3)
+F1 = F*1/z
+G1 = partfrac(F1)
+Gs = expand(G1*z)
+```
+
+%% Output
+
+    F = (sym)
    
          6⋅z
      ────────────
       2
      z  - 4⋅z + 3
    
-    F2 = (sym)
+    F1 = (sym)
    
-          3       9
+           6
+      ────────────
+       2
+      z  - 4⋅z + 3
+    
+    G1 = (sym)
+    
+          3       3
+      - ───── + ─────
+        z - 1   z - 3
+    
+    Gs = (sym)
+    
+         3⋅z     3⋅z
      - ───── + ─────
        z - 1   z - 3
    

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">

 Per Definition ist dies auch die Impulsantwort der Übertragungsfunktion, was wir wiederum mit der Funktion <code>impulse</code> aus Matlab/Octave überprüfen können.

 %% Cell type:code id: tags:

 ``` octave
 x2 = impulse(G)
 plot(k,x2(1:5))
 ```

 %% Output

    x2 =
    
         0
         6
        24
        78
       240
       726
    



 %% Cell type:code id: tags:

 ``` octave
 ```
-
-%% Cell type:code id: tags:
-
-``` octave
-```