add V6-V9

20e5fa06 · zhiyupan · 2998eb23 · 20e5fa06 · 20e5fa06 · 20e5fa06
Commit 20e5fa06 authored 3 years ago by zhiyupan
--- a/notebooks/alt/Octsim/@DTStateSpace/DTStateSpace.m
+++ b/notebooks/alt/Octsim/@DTStateSpace/DTStateSpace.m
@@ -7,22 +7,6 @@ classdef DTStateSpace < DTBlock
        xo 
    end
    methods
-        function c = DTStateSpace(in,out,A,B,C,D,T)
-            c = c@DTBlock();
-            c.A = A;
-            c.B = B;
-            c.C = C;
-            c.D = D;
-            c.Ts = T;
-            c.ninput = size(B,2);
-            c.noutput = size(C,1);
-            c.inpos = in;
-            c.outpos = out;
-            c.nstate = size(c.A,1);
-            c.xo = zeros(1,c.nstate);
-            c.x = c.xo;
-        end 
        function c = DTStateSpace(in,out,A,B,C,D,T,xo)
            c = c@DTBlock();
            c.A = A;

--- a/notebooks/alt/Octsim/@myss2tf2/myss2tf2.m
+++ b/notebooks/alt/Octsim/@myss2tf2/myss2tf2.m
+function [num den] = myss2tf2(A,B,C)
+n2 =  -A(1,1)*B(2)*C(2)+A(1,2)*B(2)*C(1)+A(2,1)*B(1)*C(2)-A(2,2)*B(1)*C(1); 
+n1 =  B(1)*C(1)+B(2)*C(2);
+d1 =  -A(1,1)-A(2,2);
+d2 =  A(1,1)*A(2,2)-A(1,2)*A(2,1);
+num = [n1 n2];
+den = [1 d1 d2];
--- a/notebooks/exam example/Bilder/Turbine.png
+++ b/notebooks/exam example/Bilder/Turbine.png
--- a/notebooks/exam example/Bilder/TurbineSchema.png
+++ b/notebooks/exam example/Bilder/TurbineSchema.png
--- a/notebooks/exam example/V6.ipynb
+++ b/notebooks/exam example/V6.ipynb
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9653081e-df65-4ccd-9204-e11c542a469f",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Gegeben ist die Prinzipskizze einer Wasserturbine zur Stromerzeugung:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "880aa4ff-2004-4875-affc-5759aba8104e",
+   "metadata": {},
+   "source": [
+    "<img src=\"Bilder/Turbine.png\" alt=\"drawing\" width=\"600\"  height=\"300\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a27e62db-897e-41e8-9b56-fd8c6b48f3bf",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Die Wasserturbine sei in dieser Aufgabe die zu betrachtende Regelstrecke, die als linear\n",
+    "angenommen wird. Damit lässt sich das System der Wasserturbine gemäß des nachfolgenden\n",
+    "Wirkungsplans darstellen:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "add7e04a-83e6-4685-b845-47137f44a388",
+   "metadata": {},
+   "source": [
+    "<img src=\"Bilder/TurbineSchema.png\" alt=\"drawing\" width=\"600\"  height=\"300\"/>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "e050a273-a079-40b9-ac0c-194e5e7f12e8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "% Necessary to use control toolbox\n",
+    "pkg load control\n",
+    "clear all"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1943a7dc-b837-4623-8792-a1775026155d",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Geben Sie die Zustandsraumdarstellung des Systems"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "ef3d3be5-73cd-48d2-a773-d177b638230a",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Kb =  500\n",
+      "Kt =  104\n",
+      "Kv =  2\n",
+      "J =  100\n"
+     ]
+    }
+   ],
+   "source": [
+    "Kb = 500\n",
+    "Kt = 104\n",
+    "Kv = 2\n",
+    "J = 100"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "402f525f-8a64-4584-8d05-dbd5e42e5516",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   0.00000  -2.00000\n",
+      "   1.04000  -5.00000\n",
+      "\n",
+      "B =\n",
+      "\n",
+      "   1\n",
+      "   0\n",
+      "\n",
+      "C =\n",
+      "\n",
+      "   0   1\n",
+      "\n",
+      "D = 0\n"
+     ]
+    }
+   ],
+   "source": [
+    "A = [0 -Kv;Kt/J -Kb/J]\n",
+    "B = [1; 0]\n",
+    "C = [0 1]\n",
+    "D = 0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8629e5a2-ae35-418d-8a65-7e511440917e",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Dann berechnen wir die Eigenwerte "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "1ed03ed5-3913-4107-95c9-f48e52bb50ee",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "p =\n",
+      "\n",
+      "  -4.54206\n",
+      "  -0.45794\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "p = eigs(A)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0353e887-ebe2-44d0-b7fb-a191622cabff",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Dann wenden wir das Caley-Hamilton-Theorem an"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "0d90c253-316f-47ed-9f7b-561ff0ed4868",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Ac =\n",
+      "\n",
+      "   1.00000  -4.54206\n",
+      "   1.00000  -0.45794\n",
+      "\n",
+      "Bc =\n",
+      "\n",
+      "   0.63495\n",
+      "   0.95524\n",
+      "\n",
+      "alfa =\n",
+      "\n",
+      "   0.991151\n",
+      "   0.078422\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Ts = 0.1;\n",
+    "Ac = [1 p(1);1 p(2)]\n",
+    "Bc = [expm(p(1)*Ts);expm(p(2)*Ts)]\n",
+    "alfa = inv(Ac)*Bc"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d4fac333-e43d-49c7-aaec-6168b86bd6a6",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "und dann:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "828217c8-bd90-42da-8b36-9e1f05727a24",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Fi =\n",
+      "\n",
+      "   0.991151  -0.156845\n",
+      "   0.081559   0.599039\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Fi = alfa(1)*eye(2)+alfa(2)*A"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6ed8d3e-8d76-451c-8480-7182a8723ab2",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Wir Konnen vergleichen unsere Berechnungen mit Octave:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "058ad783-d553-4bcc-873c-a6f521c16975",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Fio =\n",
+      "\n",
+      "   0.991151  -0.156845\n",
+      "   0.081559   0.599039\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Fio = expm(A*Ts)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "63fb7dec-c4e2-4e53-b6f9-bea0649fd521",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Die andere Matrix können wir so berechnen:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "0c2d94f8-536c-421a-98ae-69771c0e6ca2",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Gi =\n",
+      "\n",
+      "   0.0996930\n",
+      "   0.0044243\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Gi = (Fi-eye(2))*inv(A)*B"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8a4be558-f51c-415e-96f7-922b8e417422",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Zu vergleichen:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "69e7fd06-f71a-48ec-af34-d2e406f9067c",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "sys.a =\n",
+      "         x1    x2\n",
+      "   x1     0    -2\n",
+      "   x2  1.04    -5\n",
+      "\n",
+      "sys.b =\n",
+      "       u1\n",
+      "   x1   1\n",
+      "   x2   0\n",
+      "\n",
+      "sys.c =\n",
+      "       x1  x2\n",
+      "   y1   0   1\n",
+      "\n",
+      "sys.d =\n",
+      "       u1\n",
+      "   y1   0\n",
+      "\n",
+      "Continuous-time model.\n",
+      "\n",
+      "sysd.a =\n",
+      "            x1       x2\n",
+      "   x1   0.9912  -0.1568\n",
+      "   x2  0.08156    0.599\n",
+      "\n",
+      "sysd.b =\n",
+      "             u1\n",
+      "   x1   0.09969\n",
+      "   x2  0.004424\n",
+      "\n",
+      "sysd.c =\n",
+      "       x1  x2\n",
+      "   y1   0   1\n",
+      "\n",
+      "sysd.d =\n",
+      "       u1\n",
+      "   y1   0\n",
+      "\n",
+      "Sampling time: 0.1 s\n",
+      "Discrete-time model.\n"
+     ]
+    }
+   ],
+   "source": [
+    "sys = ss(A,B,C,D)\n",
+    "sysd = c2d(sys,Ts,'zoh')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4fb92a07-3450-4c37-bef4-7e3734cc8c20",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Jetzt können wir die Übertragungsfunktion in z-domän berechnen:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "cbf41502-116d-40b5-8fec-39faf2062a03",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "num =\n",
+      "\n",
+      "   0.0044243   0.0037458\n",
+      "\n",
+      "den =\n",
+      "\n",
+      "   1.00000  -1.59019   0.60653\n",
+      "\n",
+      "\n",
+      "Transfer function 'Gz' from input 'u1' to output ...\n",
+      "\n",
+      "      0.004424 z + 0.003746\n",
+      " y1:  ---------------------\n",
+      "      z^2 - 1.59 z + 0.6065\n",
+      "\n",
+      "Sampling time: 0.1 s\n",
+      "Discrete-time model.\n"
+     ]
+    }
+   ],
+   "source": [
+    "[num den] = myss2tf2(Fi,Gi,C)\n",
+    "Gz = tf(num,den,Ts)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6ebd4c8d-4218-41b7-aa86-902033541283",
+   "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:9653081e-df65-4ccd-9204-e11c542a469f tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Gegeben ist die Prinzipskizze einer Wasserturbine zur Stromerzeugung:
+%% Cell type:markdown id:880aa4ff-2004-4875-affc-5759aba8104e tags:
+<img src="Bilder/Turbine.png" alt="drawing" width="600"  height="300"/>
+%% Cell type:markdown id:a27e62db-897e-41e8-9b56-fd8c6b48f3bf tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Die Wasserturbine sei in dieser Aufgabe die zu betrachtende Regelstrecke, die als linear
+angenommen wird. Damit lässt sich das System der Wasserturbine gemäß des nachfolgenden
+Wirkungsplans darstellen:
+%% Cell type:markdown id:add7e04a-83e6-4685-b845-47137f44a388 tags:
+<img src="Bilder/TurbineSchema.png" alt="drawing" width="600"  height="300"/>
+%% Cell type:code id:e050a273-a079-40b9-ac0c-194e5e7f12e8 tags:
+``` octave
+% Necessary to use control toolbox
+pkg load control
+clear all
+```
+%% Cell type:markdown id:1943a7dc-b837-4623-8792-a1775026155d tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Geben Sie die Zustandsraumdarstellung des Systems
+%% Cell type:code id:ef3d3be5-73cd-48d2-a773-d177b638230a tags:
+``` octave
+Kb = 500
+Kt = 104
+Kv = 2
+J = 100
+```
+%% Output
+    Kb =  500
+    Kt =  104
+    Kv =  2
+    J =  100
+%% Cell type:code id:402f525f-8a64-4584-8d05-dbd5e42e5516 tags:
+``` octave
+A = [0 -Kv;Kt/J -Kb/J]
+B = [1; 0]
+C = [0 1]
+D = 0
+```
+%% Output
+    A =
+       0.00000  -2.00000
+       1.04000  -5.00000
+    B =
+       1
+       0
+    C =
+       0   1
+    D = 0
+%% Cell type:markdown id:8629e5a2-ae35-418d-8a65-7e511440917e tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Dann berechnen wir die Eigenwerte
+%% Cell type:code id:1ed03ed5-3913-4107-95c9-f48e52bb50ee tags:
+``` octave
+p = eigs(A)
+```
+%% Output
+    p =
+      -4.54206
+      -0.45794
+%% Cell type:markdown id:0353e887-ebe2-44d0-b7fb-a191622cabff tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Dann wenden wir das Caley-Hamilton-Theorem an
+%% Cell type:code id:0d90c253-316f-47ed-9f7b-561ff0ed4868 tags:
+``` octave
+Ts = 0.1;
+Ac = [1 p(1);1 p(2)]
+Bc = [expm(p(1)*Ts);expm(p(2)*Ts)]
+alfa = inv(Ac)*Bc
+```
+%% Output
+    Ac =
+       1.00000  -4.54206
+       1.00000  -0.45794
+    Bc =
+       0.63495
+       0.95524
+    alfa =
+       0.991151
+       0.078422
+%% Cell type:markdown id:d4fac333-e43d-49c7-aaec-6168b86bd6a6 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+und dann:
+%% Cell type:code id:828217c8-bd90-42da-8b36-9e1f05727a24 tags:
+``` octave
+Fi = alfa(1)*eye(2)+alfa(2)*A
+```
+%% Output
+    Fi =
+       0.991151  -0.156845
+       0.081559   0.599039
+%% Cell type:markdown id:a6ed8d3e-8d76-451c-8480-7182a8723ab2 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Wir Konnen vergleichen unsere Berechnungen mit Octave:
+%% Cell type:code id:058ad783-d553-4bcc-873c-a6f521c16975 tags:
+``` octave
+Fio = expm(A*Ts)
+```
+%% Output
+    Fio =
+       0.991151  -0.156845
+       0.081559   0.599039
+%% Cell type:markdown id:63fb7dec-c4e2-4e53-b6f9-bea0649fd521 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Die andere Matrix können wir so berechnen:
+%% Cell type:code id:0c2d94f8-536c-421a-98ae-69771c0e6ca2 tags:
+``` octave
+Gi = (Fi-eye(2))*inv(A)*B
+```
+%% Output
+    Gi =
+       0.0996930
+       0.0044243
+%% Cell type:markdown id:8a4be558-f51c-415e-96f7-922b8e417422 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Zu vergleichen:
+%% Cell type:code id:69e7fd06-f71a-48ec-af34-d2e406f9067c tags:
+``` octave
+sys = ss(A,B,C,D)
+sysd = c2d(sys,Ts,'zoh')
+```
+%% Output
+    sys.a =
+             x1    x2
+       x1     0    -2
+       x2  1.04    -5
+    sys.b =
+           u1
+       x1   1
+       x2   0
+    sys.c =
+           x1  x2
+       y1   0   1
+    sys.d =
+           u1
+       y1   0
+    Continuous-time model.
+    sysd.a =
+                x1       x2
+       x1   0.9912  -0.1568
+       x2  0.08156    0.599
+    sysd.b =
+                 u1
+       x1   0.09969
+       x2  0.004424
+    sysd.c =
+           x1  x2
+       y1   0   1
+    sysd.d =
+           u1
+       y1   0
+    Sampling time: 0.1 s
+    Discrete-time model.
+%% Cell type:markdown id:4fb92a07-3450-4c37-bef4-7e3734cc8c20 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Jetzt können wir die Übertragungsfunktion in z-domän berechnen:
+%% Cell type:code id:cbf41502-116d-40b5-8fec-39faf2062a03 tags:
+``` octave
+[num den] = myss2tf2(Fi,Gi,C)
+Gz = tf(num,den,Ts)
+```
+%% Output
+    num =
+       0.0044243   0.0037458
+    den =
+       1.00000  -1.59019   0.60653
+    Transfer function 'Gz' from input 'u1' to output ...
+          0.004424 z + 0.003746
+     y1:  ---------------------
+          z^2 - 1.59 z + 0.6065
+    Sampling time: 0.1 s
+    Discrete-time model.
+%% Cell type:code id:6ebd4c8d-4218-41b7-aa86-902033541283 tags:
+``` octave
+```
--- a/notebooks/exam example/V7.ipynb
+++ b/notebooks/exam example/V7.ipynb
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "c47fa3d9-51c0-4181-89d4-42bc55f053c1",
+   "metadata": {},
+   "source": [
+    "# <span style='color:OrangeRed'>V7 STEUERBARKEIT UND BEOBACHTBARKEIT</span>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9653081e-df65-4ccd-9204-e11c542a469f",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Ziel ist hier überzuprufen Sie, ob jedes lineare, zeitdiskrete System 3. Ordnung in Regelungsnormalform/Beobachtungsnormalform\n",
+    "steuerbar bzw. beobachtbar ist.</p>\n",
+    "b1) Stellen Sie für ein beliebiges lineares, zeitdiskretes System 3. Ordnung das Zustandsraummodell\n",
+    "in Beobachtungsnormalform auf.</p>\n",
+    "b2) Stellen Sie die Beobachtbarkeitsmatrix für das System aus Aufgabenteil b1) auf und\n",
+    "begründen Sie, ob jenes System stets beobachtbar ist.</p>\n",
+    "b3) Verifizieren Sie, dass die Steuerbarkeitsmatrix eines beliebigen linearen, zeitdiskreten\n",
+    "Systems 3. Ordnung in Regelungsnormalform der Beobachtbarkeitsmatrix aus Aufgabenteil\n",
+    "b2) entspricht und begründen Sie damit, ob jedes lineare, zeitdiskrete\n",
+    "System 3. Ordnung in Regelungsnormalform stets steuerbar ist.</p>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "1cf4028f-c9ee-468a-995a-2284567c6a02",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "pkg load symbolic"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a27e62db-897e-41e8-9b56-fd8c6b48f3bf",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Die Zustandsraumdarstellung in Beobachtungsnormalform für ein beliebiges lineares,\n",
+    "zeitdiskretes System 3. Ordnung lautet:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "e050a273-a079-40b9-ac0c-194e5e7f12e8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "syms a1 a2 a3\n",
+    "syms b0 b1 b2 b3"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "402f525f-8a64-4584-8d05-dbd5e42e5516",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A = (sym 3×3 matrix)\n",
+      "\n",
+      "  ⎡0  0  -a₃⎤\n",
+      "  ⎢         ⎥\n",
+      "  ⎢1  0  -a₂⎥\n",
+      "  ⎢         ⎥\n",
+      "  ⎣0  1  -a₁⎦\n",
+      "\n",
+      "B = (sym 3×1 matrix)\n",
+      "\n",
+      "  ⎡-a₃⋅b₀ + b₃⎤\n",
+      "  ⎢           ⎥\n",
+      "  ⎢-a₂⋅b₀ + b₂⎥\n",
+      "  ⎢           ⎥\n",
+      "  ⎣-a₁⋅b₀ + b₁⎦\n",
+      "\n",
+      "C =\n",
+      "\n",
+      "   0   0   1\n",
+      "\n",
+      "D = (sym) b₀\n"
+     ]
+    }
+   ],
+   "source": [
+    "A = [0 0 -a3; 1 0 -a2; 0 1 -a1]\n",
+    "B = [b3-b0*a3; b2-b0*a2; b1-b0*a1]\n",
+    "C = [0 0 1]\n",
+    "D = b0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8629e5a2-ae35-418d-8a65-7e511440917e",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Aufstellen der Beobachtbarkeitsmatrix"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "1ed03ed5-3913-4107-95c9-f48e52bb50ee",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "So = (sym 3×3 matrix)\n",
+      "\n",
+      "  ⎡0   0      1    ⎤\n",
+      "  ⎢                ⎥\n",
+      "  ⎢0   1     -a₁   ⎥\n",
+      "  ⎢                ⎥\n",
+      "  ⎢          2     ⎥\n",
+      "  ⎣1  -a₁  a₁  - a₂⎦\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "So = [C; C*A; C*A*A]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0353e887-ebe2-44d0-b7fb-a191622cabff",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Um Beobachtbarkeit überzuprufen berechnnen wir die Determinant"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "0d90c253-316f-47ed-9f7b-561ff0ed4868",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans = (sym) -1\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(So)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d4fac333-e43d-49c7-aaec-6168b86bd6a6",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "So hat denn vollen Rang, weshalb jedes lineare, zeitdiskrete System 3. Ordnung in Beobachtungsnormalform\n",
+    "beobachtbar ist."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6ed8d3e-8d76-451c-8480-7182a8723ab2",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Die Regelungsnormalform und Beobachtungsnormalform sind zueinander dual.\n",
+    "Das heißt es gilt:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "828217c8-bd90-42da-8b36-9e1f05727a24",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "As = (sym 3×3 matrix)\n",
+      "\n",
+      "  ⎡ 0    1    0 ⎤\n",
+      "  ⎢             ⎥\n",
+      "  ⎢ 0    0    1 ⎥\n",
+      "  ⎢             ⎥\n",
+      "  ⎣-a₃  -a₂  -a₁⎦\n",
+      "\n",
+      "Bs =\n",
+      "\n",
+      "   0\n",
+      "   0\n",
+      "   1\n",
+      "\n",
+      "Cs = (sym) [-a₃⋅b₀ + b₃  -a₂⋅b₀ + b₂  -a₁⋅b₀ + b₁]  (1×3 matrix)\n",
+      "Ds = (sym) b₀\n"
+     ]
+    }
+   ],
+   "source": [
+    "As = A.'\n",
+    "Bs = C.'\n",
+    "Cs = B.'\n",
+    "Ds = D"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "63fb7dec-c4e2-4e53-b6f9-bea0649fd521",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Aufstellen der Steuerbarkeitsmatrix"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "0c2d94f8-536c-421a-98ae-69771c0e6ca2",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Ss = (sym 3×3 matrix)\n",
+      "\n",
+      "  ⎡0   0      1    ⎤\n",
+      "  ⎢                ⎥\n",
+      "  ⎢0   1     -a₁   ⎥\n",
+      "  ⎢                ⎥\n",
+      "  ⎢          2     ⎥\n",
+      "  ⎣1  -a₁  a₁  - a₂⎦\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Ss = [Bs As*Bs As*As*Bs]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8a4be558-f51c-415e-96f7-922b8e417422",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "Ss = So. Um Steuerbarkeit überzuprufen berechnnen wir auf jedem Fall die Determinant"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "69e7fd06-f71a-48ec-af34-d2e406f9067c",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans = (sym) -1\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(Ss)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4fb92a07-3450-4c37-bef4-7e3734cc8c20",
+   "metadata": {},
+   "source": [
+    "<div style=\"font-family: 'times'; font-size: 13pt; text-align: justify\">\n",
+    "So hat denn vollen Rang (wie erwartet), weshalb jedes lineare, zeitdiskrete System 3. Ordnung in Regelungnormalform\n",
+    "steuerbar ist."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6ebd4c8d-4218-41b7-aa86-902033541283",
+   "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:c47fa3d9-51c0-4181-89d4-42bc55f053c1 tags:
+# <span style='color:OrangeRed'>V7 STEUERBARKEIT UND BEOBACHTBARKEIT</span>
+%% Cell type:markdown id:9653081e-df65-4ccd-9204-e11c542a469f tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Ziel ist hier überzuprufen Sie, ob jedes lineare, zeitdiskrete System 3. Ordnung in Regelungsnormalform/Beobachtungsnormalform
+steuerbar bzw. beobachtbar ist.</p>
+b1) Stellen Sie für ein beliebiges lineares, zeitdiskretes System 3. Ordnung das Zustandsraummodell
+in Beobachtungsnormalform auf.</p>
+b2) Stellen Sie die Beobachtbarkeitsmatrix für das System aus Aufgabenteil b1) auf und
+begründen Sie, ob jenes System stets beobachtbar ist.</p>
+b3) Verifizieren Sie, dass die Steuerbarkeitsmatrix eines beliebigen linearen, zeitdiskreten
+Systems 3. Ordnung in Regelungsnormalform der Beobachtbarkeitsmatrix aus Aufgabenteil
+b2) entspricht und begründen Sie damit, ob jedes lineare, zeitdiskrete
+System 3. Ordnung in Regelungsnormalform stets steuerbar ist.</p>
+%% Cell type:code id:1cf4028f-c9ee-468a-995a-2284567c6a02 tags:
+``` octave
+pkg load symbolic
+```
+%% Cell type:markdown id:a27e62db-897e-41e8-9b56-fd8c6b48f3bf tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Die Zustandsraumdarstellung in Beobachtungsnormalform für ein beliebiges lineares,
+zeitdiskretes System 3. Ordnung lautet:
+%% Cell type:code id:e050a273-a079-40b9-ac0c-194e5e7f12e8 tags:
+``` octave
+syms a1 a2 a3
+syms b0 b1 b2 b3
+```
+%% Cell type:code id:402f525f-8a64-4584-8d05-dbd5e42e5516 tags:
+``` octave
+A = [0 0 -a3; 1 0 -a2; 0 1 -a1]
+B = [b3-b0*a3; b2-b0*a2; b1-b0*a1]
+C = [0 0 1]
+D = b0
+```
+%% Output
+    A = (sym 3×3 matrix)
+      ⎡0  0  -a₃⎤
+      ⎢         ⎥
+      ⎢1  0  -a₂⎥
+      ⎢         ⎥
+      ⎣0  1  -a₁⎦
+    B = (sym 3×1 matrix)
+      ⎡-a₃⋅b₀ + b₃⎤
+      ⎢           ⎥
+      ⎢-a₂⋅b₀ + b₂⎥
+      ⎢           ⎥
+      ⎣-a₁⋅b₀ + b₁⎦
+    C =
+       0   0   1
+    D = (sym) b₀
+%% Cell type:markdown id:8629e5a2-ae35-418d-8a65-7e511440917e tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Aufstellen der Beobachtbarkeitsmatrix
+%% Cell type:code id:1ed03ed5-3913-4107-95c9-f48e52bb50ee tags:
+``` octave
+So = [C; C*A; C*A*A]
+```
+%% Output
+    So = (sym 3×3 matrix)
+      ⎡0   0      1    ⎤
+      ⎢                ⎥
+      ⎢0   1     -a₁   ⎥
+      ⎢                ⎥
+      ⎢          2     ⎥
+      ⎣1  -a₁  a₁  - a₂⎦
+%% Cell type:markdown id:0353e887-ebe2-44d0-b7fb-a191622cabff tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Um Beobachtbarkeit überzuprufen berechnnen wir die Determinant
+%% Cell type:code id:0d90c253-316f-47ed-9f7b-561ff0ed4868 tags:
+``` octave
+det(So)
+```
+%% Output
+    ans = (sym) -1
+%% Cell type:markdown id:d4fac333-e43d-49c7-aaec-6168b86bd6a6 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+So hat denn vollen Rang, weshalb jedes lineare, zeitdiskrete System 3. Ordnung in Beobachtungsnormalform
+beobachtbar ist.
+%% Cell type:markdown id:a6ed8d3e-8d76-451c-8480-7182a8723ab2 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Die Regelungsnormalform und Beobachtungsnormalform sind zueinander dual.
+Das heißt es gilt:
+%% Cell type:code id:828217c8-bd90-42da-8b36-9e1f05727a24 tags:
+``` octave
+As = A.'
+Bs = C.'
+Cs = B.'
+Ds = D
+```
+%% Output
+    As = (sym 3×3 matrix)
+      ⎡ 0    1    0 ⎤
+      ⎢             ⎥
+      ⎢ 0    0    1 ⎥
+      ⎢             ⎥
+      ⎣-a₃  -a₂  -a₁⎦
+    Bs =
+       0
+       0
+       1
+    Cs = (sym) [-a₃⋅b₀ + b₃  -a₂⋅b₀ + b₂  -a₁⋅b₀ + b₁]  (1×3 matrix)
+    Ds = (sym) b₀
+%% Cell type:markdown id:63fb7dec-c4e2-4e53-b6f9-bea0649fd521 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Aufstellen der Steuerbarkeitsmatrix
+%% Cell type:code id:0c2d94f8-536c-421a-98ae-69771c0e6ca2 tags:
+``` octave
+Ss = [Bs As*Bs As*As*Bs]
+```
+%% Output
+    Ss = (sym 3×3 matrix)
+      ⎡0   0      1    ⎤
+      ⎢                ⎥
+      ⎢0   1     -a₁   ⎥
+      ⎢                ⎥
+      ⎢          2     ⎥
+      ⎣1  -a₁  a₁  - a₂⎦
+%% Cell type:markdown id:8a4be558-f51c-415e-96f7-922b8e417422 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+Ss = So. Um Steuerbarkeit überzuprufen berechnnen wir auf jedem Fall die Determinant
+%% Cell type:code id:69e7fd06-f71a-48ec-af34-d2e406f9067c tags:
+``` octave
+det(Ss)
+```
+%% Output
+    ans = (sym) -1
+%% Cell type:markdown id:4fb92a07-3450-4c37-bef4-7e3734cc8c20 tags:
+<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
+So hat denn vollen Rang (wie erwartet), weshalb jedes lineare, zeitdiskrete System 3. Ordnung in Regelungnormalform
+steuerbar ist.
+%% Cell type:code id:6ebd4c8d-4218-41b7-aa86-902033541283 tags:
+``` octave
+```
--- a/notebooks/exam example/V8.ipynb
+++ b/notebooks/exam example/V8.ipynb
--- a/notebooks/exam example/V9.ipynb
+++ b/notebooks/exam example/V9.ipynb