add notebook V03.2.ipynb

d536a621 · Sebastian Schwarz · 730e9c13 · d536a621 · d536a621
Commit d536a621 authored 8 months ago by Sebastian Schwarz
--- a/sys2-jupyter-notebooks/exam_examples/V03.2.ipynb
+++ b/sys2-jupyter-notebooks/exam_examples/V03.2.ipynb
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "4e14da09-e18a-4a73-a754-5514c4046d1c",
+   "metadata": {},
+   "source": [
+    "# <span style='color:OrangeRed'>V3 STABILITÄT DISKRETER SYSTEME TEIL 2</span>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33f7d33e-5729-4ee8-8fc5-8f97428f7b35",
+   "metadata": {},
+   "source": [
+    "Betrachtet wird der in den nachfolgenden Abbildung dargestellte Abstastregelkreis bestehend aus einer zeitkontinuierlichen Regelstrecke Gs(s), einem Halteglied nullter Ordnung Ho(s) und einem zeitdiskreten Regler Gr(z)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9dc0ece7-9602-45c5-b55d-f2e5acfd2e00",
+   "metadata": {},
+   "source": [
+    "<img src=\"figures/SampledControlLoop.jpeg\" alt=\"drawing\" width=\"600\"  height=\"300\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e24e18ab-6d8c-4500-8f12-aa5b6629e9f4",
+   "metadata": {},
+   "source": [
+    "Die Gesamtübertragungsfunktion Gges(z) des geschlossenes Regelkreis mit Abtastzeit Ts = 0.1sec lautet:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "id": "4f8424b7-b29a-4dcc-85e0-1e00fe14c2a6",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Z =\n",
+      "\n",
+      "   0.20000  -0.80000  -0.90000   0.00000\n",
+      "\n",
+      "N =\n",
+      "\n",
+      "   1.00000  -1.70000   0.80000   0.50000  -0.50000\n",
+      "\n",
+      "\n",
+      "Transfer function 'G' from input 'u1' to output ...\n",
+      "\n",
+      "            0.2 z^3 - 0.8 z^2 - 0.9 z      \n",
+      " y1:  -------------------------------------\n",
+      "      z^4 - 1.7 z^3 + 0.8 z^2 + 0.5 z - 0.5\n",
+      "\n",
+      "Sampling time: 0.1 s\n",
+      "Discrete-time model.\n"
+     ]
+    }
+   ],
+   "source": [
+    "pkg load control\n",
+    "Z = [0.2 -0.8 -0.9 0]\n",
+    "N = [1 -1.7 0.8 0.5 -0.5]\n",
+    "G = tf(Z,N,0.1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9b66b0b0-767e-4140-83d3-a9017969e1b2",
+   "metadata": {},
+   "source": [
+    "Es soll die Stabilität des geschlossenen Regelkreises unteresucht werden.\n",
+    "Wenden Sie zu diesem Zweck das algebraische Stabilitätskriterium nach Jury an.\n",
+    "\n",
+    "Prüfung der notwendigen Bedingung:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "id": "195ef940-7487-4010-8dff-56b74852c8b7",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "T1 =  0.10000\n",
+      "T2 =  2.5000\n"
+     ]
+    }
+   ],
+   "source": [
+    "T1 = polyval(N,1)\n",
+    "T2 = polyval(N,-1)*(-1)^(length(N)-1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "542aa21b-775e-4018-8265-66e8250e7f58",
+   "metadata": {},
+   "source": [
+    "T1 und T2 müssen größer als 0 sein.\n",
+    "\n",
+    "Prüfung der hinreichenden Bedingungen:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "id": "32221e0c-874a-4791-b21e-ef736934c7cc",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "R1 =\n",
+      "\n",
+      "  -0.50000   0.50000   0.80000  -1.70000   1.00000\n",
+      "\n",
+      "R2 =\n",
+      "\n",
+      "   1.00000  -1.70000   0.80000   0.50000  -0.50000\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "R1 = [N(5) N(4) N(3) N(2) N(1)]\n",
+    "R2 = [N(1) N(2) N(3) N(4) N(5)]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "id": "3bb8fef7-d016-487f-b599-43d8378dc10c",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "b0 = -0.75000\n",
+      "b1 =  1.4500\n",
+      "b2 = -1.2000\n",
+      "b3 =  0.35000\n"
+     ]
+    }
+   ],
+   "source": [
+    "b0 = R1(1)*R1(1)-R1(5)*R1(5)\n",
+    "b1 = R1(1)*R1(2)-R1(5)*R1(4)\n",
+    "b2 = R1(1)*R1(3)-R1(5)*R1(3)\n",
+    "b3 = R1(1)*R1(4)-R1(5)*R1(2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "id": "9427eb80-a325-44cc-a871-159ebe4bbd4d",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "R3 =\n",
+      "\n",
+      "  -0.75000   1.45000  -1.20000   0.35000\n",
+      "\n",
+      "R4 =\n",
+      "\n",
+      "   0.35000  -1.20000   1.45000  -0.75000\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "R3 = [b0 b1 b2 b3]\n",
+    "R4 = [b3 b2 b1 b0]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "id": "a39447bd-d38a-43e1-adc8-68060eda571a",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "c0 =  0.44000\n",
+      "c1 = -0.66750\n",
+      "c2 =  0.39250\n"
+     ]
+    }
+   ],
+   "source": [
+    "c0 = R3(1)*R3(1)-R3(4)*R3(4)\n",
+    "c1 = R3(1)*R3(2)-R3(4)*R3(3)\n",
+    "c2 = R3(1)*R3(3)-R3(4)*R3(2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "id": "ee8b51ef-9da2-4bec-9c40-e30f81bfeaf3",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "R5 =\n",
+      "\n",
+      "   0.44000  -0.66750   0.39250\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "R5 = [c0 c1 c2]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2d41a11a-f281-4a07-9687-06d0eee47328",
+   "metadata": {},
+   "source": [
+    "Die zu überprüfenden Bedigungen sind\n",
+    "abs(R1(1))<abs(R1(5)), \n",
+    "abs(R3(1))>abs(R3(4)), \n",
+    "abs(R5(1))>abs(R5(3)). \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7529de03-0e61-425b-b2ad-cdec6b9c2260",
+   "metadata": {},
+   "source": [
+    "Wie können das auch überprufen. Wir berechnen die Nullstellen von N und betrachten deren Betrag:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "id": "7c6efadd-827e-4089-abaa-3f285e6e4a23",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "p =\n",
+      "\n",
+      "  -0.60477 + 0.00000i\n",
+      "   0.71290 + 0.65756i\n",
+      "   0.71290 - 0.65756i\n",
+      "   0.87896 + 0.00000i\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   0.60477\n",
+      "   0.96986\n",
+      "   0.96986\n",
+      "   0.87896\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "p = roots(N)\n",
+    "abs(p)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2617cdc0-e2fc-4951-9d4b-5cab91647dc4",
+   "metadata": {},
+   "source": [
+    "Die Beträge müssen kleiner als 1 sein."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "05064e3d-36a8-46ae-9661-dcbaddf31a95",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Octave",
+   "language": "octave",
+   "name": "octave"
+  },
+  "language_info": {
+   "file_extension": ".m",
+   "help_links": [
+    {
+     "text": "GNU Octave",
+     "url": "https://www.gnu.org/software/octave/support.html"
+    },
+    {
+     "text": "Octave Kernel",
+     "url": "https://github.com/Calysto/octave_kernel"
+    },
+    {
+     "text": "MetaKernel Magics",
+     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
+    }
+   ],
+   "mimetype": "text/x-octave",
+   "name": "octave",
+   "version": "5.2.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
+%% Cell type:markdown id:4e14da09-e18a-4a73-a754-5514c4046d1c tags:
+
+# <span style='color:OrangeRed'>V3 STABILITÄT DISKRETER SYSTEME TEIL 2</span>
+
+%% Cell type:markdown id:33f7d33e-5729-4ee8-8fc5-8f97428f7b35 tags:
+
+Betrachtet wird der in den nachfolgenden Abbildung dargestellte Abstastregelkreis bestehend aus einer zeitkontinuierlichen Regelstrecke Gs(s), einem Halteglied nullter Ordnung Ho(s) und einem zeitdiskreten Regler Gr(z).
+
+%% Cell type:markdown id:9dc0ece7-9602-45c5-b55d-f2e5acfd2e00 tags:
+
+<img src="figures/SampledControlLoop.jpeg" alt="drawing" width="600"  height="300"/>
+
+%% Cell type:markdown id:e24e18ab-6d8c-4500-8f12-aa5b6629e9f4 tags:
+
+Die Gesamtübertragungsfunktion Gges(z) des geschlossenes Regelkreis mit Abtastzeit Ts = 0.1sec lautet:
+
+%% Cell type:code id:4f8424b7-b29a-4dcc-85e0-1e00fe14c2a6 tags:
+
+``` octave
+pkg load control
+Z = [0.2 -0.8 -0.9 0]
+N = [1 -1.7 0.8 0.5 -0.5]
+G = tf(Z,N,0.1)
+```
+
+%% Output
+
+    Z =
+    
+       0.20000  -0.80000  -0.90000   0.00000
+    
+    N =
+    
+       1.00000  -1.70000   0.80000   0.50000  -0.50000
+    
+    
+    Transfer function 'G' from input 'u1' to output ...
+    
+                0.2 z^3 - 0.8 z^2 - 0.9 z
+     y1:  -------------------------------------
+          z^4 - 1.7 z^3 + 0.8 z^2 + 0.5 z - 0.5
+    
+    Sampling time: 0.1 s
+    Discrete-time model.
+
+%% Cell type:markdown id:9b66b0b0-767e-4140-83d3-a9017969e1b2 tags:
+
+Es soll die Stabilität des geschlossenen Regelkreises unteresucht werden.
+Wenden Sie zu diesem Zweck das algebraische Stabilitätskriterium nach Jury an.
+
+Prüfung der notwendigen Bedingung:
+
+%% Cell type:code id:195ef940-7487-4010-8dff-56b74852c8b7 tags:
+
+``` octave
+T1 = polyval(N,1)
+T2 = polyval(N,-1)*(-1)^(length(N)-1)
+```
+
+%% Output
+
+    T1 =  0.10000
+    T2 =  2.5000
+
+%% Cell type:markdown id:542aa21b-775e-4018-8265-66e8250e7f58 tags:
+
+T1 und T2 müssen größer als 0 sein.
+
+Prüfung der hinreichenden Bedingungen:
+
+%% Cell type:code id:32221e0c-874a-4791-b21e-ef736934c7cc tags:
+
+``` octave
+R1 = [N(5) N(4) N(3) N(2) N(1)]
+R2 = [N(1) N(2) N(3) N(4) N(5)]
+```
+
+%% Output
+
+    R1 =
+    
+      -0.50000   0.50000   0.80000  -1.70000   1.00000
+    
+    R2 =
+    
+       1.00000  -1.70000   0.80000   0.50000  -0.50000
+    
+
+%% Cell type:code id:3bb8fef7-d016-487f-b599-43d8378dc10c tags:
+
+``` octave
+b0 = R1(1)*R1(1)-R1(5)*R1(5)
+b1 = R1(1)*R1(2)-R1(5)*R1(4)
+b2 = R1(1)*R1(3)-R1(5)*R1(3)
+b3 = R1(1)*R1(4)-R1(5)*R1(2)
+```
+
+%% Output
+
+    b0 = -0.75000
+    b1 =  1.4500
+    b2 = -1.2000
+    b3 =  0.35000
+
+%% Cell type:code id:9427eb80-a325-44cc-a871-159ebe4bbd4d tags:
+
+``` octave
+R3 = [b0 b1 b2 b3]
+R4 = [b3 b2 b1 b0]
+```
+
+%% Output
+
+    R3 =
+    
+      -0.75000   1.45000  -1.20000   0.35000
+    
+    R4 =
+    
+       0.35000  -1.20000   1.45000  -0.75000
+    
+
+%% Cell type:code id:a39447bd-d38a-43e1-adc8-68060eda571a tags:
+
+``` octave
+c0 = R3(1)*R3(1)-R3(4)*R3(4)
+c1 = R3(1)*R3(2)-R3(4)*R3(3)
+c2 = R3(1)*R3(3)-R3(4)*R3(2)
+```
+
+%% Output
+
+    c0 =  0.44000
+    c1 = -0.66750
+    c2 =  0.39250
+
+%% Cell type:code id:ee8b51ef-9da2-4bec-9c40-e30f81bfeaf3 tags:
+
+``` octave
+R5 = [c0 c1 c2]
+```
+
+%% Output
+
+    R5 =
+    
+       0.44000  -0.66750   0.39250
+    
+
+%% Cell type:markdown id:2d41a11a-f281-4a07-9687-06d0eee47328 tags:
+
+Die zu überprüfenden Bedigungen sind
+abs(R1(1))<abs(R1(5)),
+abs(R3(1))>abs(R3(4)),
+abs(R5(1))>abs(R5(3)).
+
+%% Cell type:markdown id:7529de03-0e61-425b-b2ad-cdec6b9c2260 tags:
+
+Wie können das auch überprufen. Wir berechnen die Nullstellen von N und betrachten deren Betrag:
+
+%% Cell type:code id:7c6efadd-827e-4089-abaa-3f285e6e4a23 tags:
+
+``` octave
+p = roots(N)
+abs(p)
+```
+
+%% Output
+
+    p =
+    
+      -0.60477 + 0.00000i
+       0.71290 + 0.65756i
+       0.71290 - 0.65756i
+       0.87896 + 0.00000i
+    
+    ans =
+    
+       0.60477
+       0.96986
+       0.96986
+       0.87896
+    
+
+%% Cell type:markdown id:2617cdc0-e2fc-4951-9d4b-5cab91647dc4 tags:
+
+Die Beträge müssen kleiner als 1 sein.
+
+%% Cell type:code id:05064e3d-36a8-46ae-9661-dcbaddf31a95 tags:
+
+``` octave
+```
--- a/sys2-jupyter-notebooks/exam_examples/figures/SampledControlLoop.jpeg
+++ b/sys2-jupyter-notebooks/exam_examples/figures/SampledControlLoop.jpeg