add V06.2.ipynb

sys2-jupyter-notebooks/exam_examples/V06.2.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "1f0f7673-ab1a-413f-b73c-0a248ba39ed9",
+   "metadata": {},
+   "source": [
+    "# <span style='color:OrangeRed'>V6 NORMALFORMEN IN ZUSTANDRAUMDARSTELLUNG TEIL 2</span>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9653081e-df65-4ccd-9204-e11c542a469f",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Es ist ein lineares System gegeben, das im Zustandsraum folgende Beschreibung hat:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "825956d9-8873-4d83-bf39-7c8d40da2aff",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "a =  5\n",
+      "b =  1\n",
+      "c =  2\n",
+      "A =\n",
+      "\n",
+      "   2   0\n",
+      "   1   1\n",
+      "\n",
+      "B =\n",
+      "\n",
+      "   1\n",
+      "   5\n",
+      "\n",
+      "C =\n",
+      "\n",
+      "   0   1\n",
+      "\n",
+      "D = 0\n"
+     ]
+    }
+   ],
+   "source": [
+    "clear all\n",
+    "pkg load symbolic\n",
+    "\n",
+    "a = 5\n",
+    "b = 1\n",
+    "c = 2\n",
+    "\n",
+    "A = [c 0;1 b]\n",
+    "B = [b; a]\n",
+    "C = [0 1]\n",
+    "D = 0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a27e62db-897e-41e8-9b56-fd8c6b48f3bf",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Finden Sie eine Transformation, die eine diagonale Form ergibt."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1943a7dc-b837-4623-8792-a1775026155d",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Zunächst müssen wir die Eigenwerte des Systems finden:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "ef3d3be5-73cd-48d2-a773-d177b638230a",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "p =\n",
+      "\n",
+      "   2\n",
+      "   1\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "p = eigs(A)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a160dc2c-e6be-4329-9a7a-69f7689a7354",
+   "metadata": {},
+   "source": [
+    "Wir müssen nun die beiden Eigenvektoren berechnen, die den beiden Eigenwerten zugeordnet sind:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "1ed03ed5-3913-4107-95c9-f48e52bb50ee",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "V =\n",
+      "\n",
+      "   0.00000   0.70711\n",
+      "   1.00000   0.70711\n",
+      "\n",
+      "D =\n",
+      "\n",
+      "Diagonal Matrix\n",
+      "\n",
+      "   1   0\n",
+      "   0   2\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "[V D]=eig(A)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5cea5a50-7f13-4d9e-b343-de64bacb84f7",
+   "metadata": {},
+   "source": [
+    "Wir können das Ergebnis auch überprufen:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "5ef87d39-8ab8-4ce0-adc2-d392311d8a9c",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "At =\n",
+      "\n",
+      "   1   0\n",
+      "   0   2\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "At = inv(V)*A*V"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e81354c4-0adf-4909-a490-32f2ee713788",
+   "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
+}