Add systheo2functions.py to notebooks/V04..., notebooks/V06..., notebooks/V07, notebooks/V08

e75740e7 · JupyterHub User · e5f742fb · e75740e7 · e75740e7 · e75740e7
Commit e75740e7 authored 3 years ago by JupyterHub User
--- a/notebooks/V04_regelung_diskreter_systeme.ipynb
+++ b/notebooks/V04_regelung_diskreter_systeme.ipynb
--- a/notebooks/V06_normalformen_in_zustandsraumdarstellung.ipynb
+++ b/notebooks/V06_normalformen_in_zustandsraumdarstellung.ipynb
--- a/notebooks/V07_steuerbarkeit_und_beobachtbarkeit.ipynb
+++ b/notebooks/V07_steuerbarkeit_und_beobachtbarkeit.ipynb
@@ -11,26 +11,12 @@
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
-    "jupyter": {
-     "source_hidden": true
-    }
+    "tags": []
   },
   "outputs": [],
   "source": [
-    "import numpy as np\n",
-    "pi = np.pi\n",
-    "sin = np.sin\n",
-    "cos = np.cos\n",
-    "sqrt = np.sqrt\n",
-    "exp = np.exp\n",
-    "atan2 = np.arctan2\n",
-    "log10 = np.log10\n",
-    "\n",
-    "def print_det(M):\n",
-    "    print(\"Determinante: \"+str(np.around(np.linalg.det(M))))\n",
-    "\n",
-    "def print_rank(M):\n",
-    "    print(\"Rang: \"+str(np.linalg.matrix_rank(M)))"
+    "from systheo2functions import *\n",
+    "%matplotlib inline"
   ]
  },
  {
@@ -301,18 +287,11 @@
    " <code>det(S_O)</code> $\\ne$ 0 und <code>rank(S_O)</code> = $n$ = $3$\n",
    "<br>Damit ist das gegebene System  <span style='color:red'>vollständig beobachtbar</span>! "
   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
@@ -326,7 +305,7 @@
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
-   "version": "3.7.6"
+   "version": "3.9.6"
  }
 },
 "nbformat": 4,

 %% Cell type:markdown id: tags:

 # <span style='color:OrangeRed'>V7 - Steuerbarkeit und Beobachtbarkeit</span>

 %% Cell type:code id: tags:

 ``` python
-import numpy as np
-pi = np.pi
-sin = np.sin
-cos = np.cos
-sqrt = np.sqrt
-exp = np.exp
-atan2 = np.arctan2
-log10 = np.log10
-
-def print_det(M):
-    print("Determinante: "+str(np.around(np.linalg.det(M))))
-
-def print_rank(M):
-    print("Rang: "+str(np.linalg.matrix_rank(M)))
+from systheo2functions import *
+%matplotlib inline
 ```

 %% Cell type:markdown id: tags:

 ## <span style='color:Gray'>Beispiel #1 </span>

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 Es wird das folgende Eingrößensystem betrachtet:
 <br>$\dot{x}(t)$= $Ax(t)$ + $bu(t)$
 <br>$y(t)$=$c^Tx(t)$
 <br><br>    $x$ =  $\left[ \begin{array}{rrrr}
           x_1   \\
           x_2   \\
           x_3   \\
           \end{array}\right] $
 <br><br>    $A$ = $\left[ \begin{array}{rrrr}
           -1 & -2 & -2 \\
            0 & -1 & 1  \\
            1 & 0  & -1 \\
          \end{array}\right] $


 %% Cell type:code id: tags:

 ``` python
 A = np.array([[-1,-1,-2],[0,-1,1],[1,0,-1]]);
 ```

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 $b$ =  $\left[ \begin{array}{rrrr}
           2   \\
           0   \\
           1   \\
          \end{array}\right] $

 %% Cell type:code id: tags:

 ``` python
 b = np.array([[2],[0],[1]]);
 ```

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 $c^T$ =  $\left[ \begin{array}{rrrr}
           1 & 1 & 0  \\
           \end{array}\right] $

 %% Cell type:code id: tags:

 ``` python
 c = np.array([1,1,0]);
 ```

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 <b>Zielsetzung</b>: Überprüfung der Steuerbarkeit und Beobachtbarkeit.
 <br>Zuerst betrachten wir die <span style='color:red'>Steuerbarkeit</span>:
 <br>Ein System 3. Ordnung ist vollständig zustandssteuerbar, wenn gilt:
 $Rang$[$B\,|\,AB\,|\,A^2B$]=$Rang$($S_C$) = $n$ = $3$
    <br><b>Mit</b>:
 <br><br>    $A$ = $\left[ \begin{array}{rrrr}
           -1 & -2 & -2 \\
            0 & -1 & 1  \\
            1 & 0  & -1 \\
          \end{array}\right] $
 <br><br>    $B$ = $b$ =  $\left[ \begin{array}{rrrr}
           2   \\
           0   \\
           1   \\
          \end{array}\right] $
    <br><b>Wird</b>:
 <br><br>    $Ab$ =  $\left[ \begin{array}{rrrr}
           -4   \\
           1  \\
           1   \\
          \end{array}\right] $
 <br><br>    $A^2b$ =  $\left[ \begin{array}{rrrr}
           0  \\
           0   \\
           -5   \\
          \end{array}\right] $
 <br>Damit erhalten wir die Steuerbarkeitsmatrix:
  <br>$Rang$[$b\,|\,Ab\,|\,A^2b$] = $\left[ \begin{array}{rrrr}
           2 & -4 & 0 \\
            0 & 1 & 0  \\
            1 & 1  & -5 \\
          \end{array}\right] $

 %% Cell type:code id: tags:

 ``` python
 column1 = b
 column2 = np.matmul(A,b)
 column3 = np.matmul(np.matmul(A,A),b)
 S_C = np.c_[column1,column2,column3]
 print("S_C:\n"+str(S_C))
 ```

 %% Output

    S_C:
    [[ 2 -4  1]
     [ 0  1  0]
     [ 1  1 -5]]

 %% Cell type:code id: tags:

 ``` python
 print_det(S_C)
 ```

 %% Output

    Determinante: -11.0

 %% Cell type:code id: tags:

 ``` python
 print_rank(S_C)
 ```

 %% Output

    Rang: 3

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 <code>det(S_C)</code> $\ne$ 0 und <code>rank(S_C)</code> = $n$ = $3$
 <br>Damit ist das gegebene System  <span style='color:red'>vollständig steuerbar</span>!

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 <br>Jetzt betrachten wir die <span style='color:red'>Beobachtbarkeit</span>:
 <br>Ein Eingrößensystem 3. Ordnung ist genau dann beobachtbar, wenn: $Rang$$\left[ \begin{array}{rrrr}
           c^T   \\
           c^TA\\
           c^TA^2\\
          \end{array}\right] $ =$Rang$($S_O$) = $n$ = $3$
 <br>Im vorliegenden Fall erhält man mit:
  $c^T$ = [$1$  $1$  $0$],  $c^TA$ = [$-1$  $-3$  $-1$], $c^TA^2$ = [$0$  $5$  $0$]
 <br> Damit erhalten wir die Beobachtbarkeitsmatrix:
 <br><br>    $S_O$ = $\left[ \begin{array}{rrrr}
           1 & 1 & 0 \\
            -1 & -3 & -1  \\
            0 & 5  & 0 \\
          \end{array}\right] $


 %% Cell type:code id: tags:

 ``` python
 S_O = np.c_[c, np.matmul(c,A), np.matmul(np.matmul(c,A),A)]
 print("S_C:\n"+str(S_O))
 ```

 %% Output

    S_C:
    [[ 1 -1  0]
     [ 1 -2  3]
     [ 0 -1  1]]

 %% Cell type:code id: tags:

 ``` python
 print_det(S_O)
 ```

 %% Output

    Determinante: 2.0

 %% Cell type:code id: tags:

 ``` python
 print_rank(S_O)
 ```

 %% Output

    Rang: 3

 %% Cell type:markdown id: tags:

 <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
 <code>det(S_O)</code> $\ne$ 0 und <code>rank(S_O)</code> = $n$ = $3$
 <br>Damit ist das gegebene System  <span style='color:red'>vollständig beobachtbar</span>!
-
-%% Cell type:code id: tags:
-
-``` python
-```

--- a/notebooks/V08_regelung_im_zustandsraum.ipynb
+++ b/notebooks/V08_regelung_im_zustandsraum.ipynb