3_HöhereRegelungstechnik.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Höhere Regelungstechnik"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aufgabe 1\n",
    "\n",
    "Gegeben ist das System\n",
    "$$ \\dot{x} = \\begin{bmatrix}a&1\\\\2&b\\end{bmatrix}x  \\ \\ \\ \\ \\ \\ \\ \\ \\ y = \\begin{bmatrix}0&1\\end{bmatrix}x \\ \\ \\ \\ \\ \\ \\ \\ mit \\ a,b \\in R $$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "Die Zustände des Systems sollen mit einem Luenberger-Beobachter mit der Matrix $ L = \\begin{bmatrix}l_1&l_2\\end{bmatrix}^T$ geschätzt werden. Die Pole des Beobachters sollen dabei bei $-2$ liegen. Geben Sie den Wert für $l_1$ in Abhängigkeit von $a$ und $b$ an.\n",
    "\n",
    "\n",
    "---\n",
    "\n",
    "Der Beobachter hat die Form:\n",
    "\n",
    "$ \\dot{\\hat{x}} = A\\hat{x} + L(y - \\hat{y}) \\\\ \\ \\ \\ = A\\hat{x} + L(Cx - C\\hat{x}) $\n",
    "\n",
    "\n",
    "wobei $ A = \\left[\\begin{array}{cc}a & 1 \\\\2 & b\\end{array}\\right]$ und $C = \\left[\\begin{array}{ll}0 & 1\\end{array}\\right]$ und die Matrix L ist gegeben durch $L = \\left[\\begin{array}{c}l_1 \\\\ l_2\\end{array}\\right]$.\n",
    "\n",
    "Die Fehlerdynamik des Beobachtersystems ist:\n",
    "\n",
    "$$\n",
    "\\dot{e} = \\dot{x} - \\dot{\\hat{x}} = Ax - (A\\hat{x} + L(Cx - C\\hat{x})) = (A - LC)e\n",
    "$$\n",
    "\n",
    "\n",
    "wobei $e = x - \\hat{x}$ der Zustandsfehler ist. Die Matrix $A - LC$ soll die Pole bei $-2$ haben, also:\n",
    "\n",
    "$$\n",
    "A - LC = \\left[\\begin{array}{cc}\n",
    "a & 1 \\\\\n",
    "2 & b\n",
    "\\end{array}\\right] - \\left[\\begin{array}{c}\n",
    "l_1 \\\\\n",
    "l_2\n",
    "\\end{array}\\right] \\left[\\begin{array}{cc}\n",
    "0 & 1\n",
    "\\end{array}\\right] = \\left[\\begin{array}{cc}\n",
    "a & 1 - l_1 \\\\\n",
    "2 & b - l_2\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "\n",
    "\n",
    "Die $ 2 \\times 2 $ Matrix soll die Wurzeln $\\lambda = -2 $ haben. Für eine $ 2 \\times 2 $ Matrix ist die charakteristische Gleichung:\n",
    "\n",
    "$$\n",
    "\\lambda^2 - \\text{Tr}(A - LC) \\lambda + \\det(A - LC) = 0\n",
    "$$\n",
    "\n",
    "$ \\text{Tr}(A - LC) = a + b - l_2 $\n",
    "$ \\det(A - LC) = a(b - l_2) - (1 - l_1)2 $\n",
    "\n",
    "- $ \\text{Tr}(A - LC) = \\lambda + \\lambda = -4 $\n",
    "- $ \\det(A - LC) = \\lambda \\times \\lambda = 4 $\n",
    "\n",
    "\n",
    "Dann haben wir:\n",
    "\n",
    "$ a + b - l_2 = -4 \\Rightarrow l_2 = a + b + 4 $\n",
    "\n",
    "und wenn wir $l_2$ in die Determinantenbedingung einsetzen:\n",
    "\n",
    "\n",
    "$ a(b - (a + b + 4)) - 2(1 - l_1) = 4 \\\\\n",
    "a(-a - 4) - 2 + 2l_1 = 4 \\\\\n",
    "2l_1 = 4 + 2 + a^2 + 4a $\n",
    "\n",
    "$l_1 = \\frac{a^2 + 4a + 6}{2}$\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aufgabe 2\n",
    "\n",
    "Gegeben sind die Übertragungsfunktionen der offenen Regelstrecken von 3 Systemen:\n",
    "\n",
    "$$ G_{0,1}(s) = \\frac{K_1}{p_1 + s} $$\n",
    "\n",
    "$$ G_{0,2}(s) = \\frac{K_2}{s(p_2 + s)} $$\n",
    "\n",
    "$$ G_{0,3}(s) = \\frac{K_3\\omega_{0}^{2}}{s^2+2D\\omega_0s+\\omega_{0}^{2}} $$\n",
    "\n",
    "\n",
    "Es gilt für die Systemparameter $D,K_1, K_2, K_3, p_1, p_2,\\omega_0>0$\n",
    "\n",
    "\n",
    "Ordnen Sie zu, welche der unten abgebildeten Amplitudenverläufe jeweils den Sensitivitätsfunktionen $S$ oder den komplementären Sensitivitätsfunktionen $T$ der oben beschriebenen Systeme entspricht.\n",
    "\n",
    "\n",
    "\n",
    "![](img/ag2.png)\n",
    "\n",
    "\n",
    "---\n",
    "\n",
    "\n",
    "$ S = \\frac{1}{1+G_0(s)} $\n",
    "\n",
    "$ T = \\frac{G_0(s)}{1+G_0(s)} $\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aufgabe 3\n",
    "\n",
    "Gegeben ist die zeitdiskrete Übertragungsfunktion\n",
    "\n",
    "$$G(z)=\\frac{1-4,5 z}{2,25 z^{2}+3 z+1}$$ \n",
    "\n",
    "\n",
    "Berechnen Sie den analytischen Ausdruck der Impulsantwort $g[k]$ für $k \\geq 1$\n",
    "\n",
    "__\n",
    "\n",
    "### die Lösung\n",
    "\n",
    "Um den analytischen Ausdruck der Impulsantwort $g[k]$ für $k≥1$ zu berechnen, müssen wir die Übertragungsfunktion $G(z)$ in den Zeitbereich transformieren. Dies geschieht mittels der inversen z-Transformation.\n",
    "\n",
    "Die z-Transformierte Funktion ist:\n",
    "\n",
    "$$ G(z) = \\frac{1 - 4.5z}{2.25z^2 + 3z + 1}$$\n",
    "\n",
    "Die Polynom im Nenner kann man faktorisieren oder die Wurzeln berechnen, um die Partialbrüche zu bestimmen. Die Partialbruchzerlegung der Funktion G(z) ergibt:\n",
    "\n",
    "$$G(z) = \\frac{-2.0}{z + \\frac{2}{3}} + \\frac{1.7778}{\\left(z + \\frac{2}{3}\\right)^2}$$\n",
    "\n",
    "\n",
    "### Inverse Z-Transformation \n",
    "\n",
    "$$\\mathcal{Z}^{-1}\\left(\\frac{1}{z - a}\\right) = a^k \\quad \\text{und} \\quad \\mathcal{Z}^{-1}\\left(\\frac{1}{(z - a)^2}\\right) = k a^k$$\n",
    "\n",
    "\n",
    "die Transformationspaare:\n",
    "\n",
    "$\\mathcal{Z}^{-1}\\left(\\frac{1}{z + a}\\right) = (-a)^k$\n",
    "\n",
    "$\\mathcal{Z}^{-1}\\left(\\frac{1}{(z + a)^2}\\right) = k (-a)^k$\n",
    "\n",
    "\n",
    "$= \\mathcal{Z}^{-1}\\left(\\frac{-2.0}{z + \\frac{2}{3}}\\right) = -2.0 \\left(-\\frac{2}{3}\\right)^k$\n",
    "\n",
    "$= \\mathcal{Z}^{-1}\\left(\\frac{1.7778}{\\left(z + \\frac{2}{3}\\right)^2}\\right) = 1.7778 k \\left(-\\frac{2}{3}\\right)^k$\n",
    "\n",
    "\n",
    "Gesamte Impulsantwort:\n",
    "\n",
    "$g[k] = -2.0 \\left(-\\frac{2}{3}\\right)^k + 1.7778 k \\left(-\\frac{2}{3}\\right)^k$\n",
    "\n",
    "$= g[k] = \\left(-\\frac{2}{3}\\right)^k \\left( -2.0 + 1.7778k \\right)$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Closed Loop Shaping"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aufgabe 4.1 Generalized Plant - Darstellung im Bildbereich\n",
    "\n",
    "Abschätzungen bleibt ein Restfehler.\n",
    "Generalized Plant: Möglichst allgemeine Darstellung von beliebigen Regelkreisen in normieler Form:\n",
    "\n",
    "$$ w \\rightarrow P \\rightarrow Z $$\n",
    "\n",
    "\n",
    "Exogene Eingänge (nicht manipulierbar):\n",
    "$$ W(t) = \\begin{bmatrix}d(t) \\\\ r(t) \\\\ n(t)\\end{bmatrix} $$\n",
    "\n",
    "Bildbereich: \n",
    "\n",
    "$$ Z(s) \\ V(s) = \\left [P(s)(W(s)\\u(s))\\right] $$\n",
    "\n",
    "$\n",
    "y = d + G.U \\\\\n",
    "z = d + g.u - r \\\\\n",
    "v = r - \\upsilon_m \\\\\n",
    "= r - y - n \\\\\n",
    "= r -n -d -G.u \n",
    "$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "Superpositionsprinzip! \\\n",
    "$ \n",
    "u = 0 \\\\\n",
    "r = 0 \\\\\n",
    "z = d Z(s) = D(s)\n",
    "$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aufgabe 4.2 Generalized Plant im Zustandsraum\n",
    "\n",
    "$$ \n",
    "\\begin{matrix}x(t) \\\\ z(t) \\\\ v(t) \\end{matrix}\n",
    "=\n",
    "\\begin{matrix}A&B_w&B_u \\\\ C_z D_(zw) & D_(zu) \\\\ C_u & D_(uw) & D_(vu)\\end{matrix}\n",
    "\n",
    "$$\n",
    "\n",
    "Zur Erinnerung: Linear Quadratic Regulator\n",
    "\n",
    "ziel: \n",
    "$ minJ  = \\[\\int_{0}^{\\infty}(x^TQx + u^TRu) \\,dx\\] $\n",
    "\n",
    "$ \\[\\int_{0^}{\\infty}(x^T (Q^(T/2))x + u^TR^(T/2)R^(1/2)u)dt \\] $\n",
    "\n",
    "$ \\[\\int__{0^}{\\infty}(z^T z) dt\\] $\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aufgabe 4.3 $H_\\infty$-Norm\n",
    "\n",
    "Gegeben ist ein Signal Input Signal Output (SISO) System\n",
    "\n",
    "$$ ||G(j\\omega)|| = sup|G(j\\omega)| $$ \n",
    "(maximale Verstärkung)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Define the parameter range\n",
    "theta_range = np.linspace(0, 2*np.pi, 100)\n",
    "\n",
    "# Initialize arrays to store J values\n",
    "J_values = []\n",
    "\n",
    "# Calculate J for each theta\n",
    "for theta in theta_range:\n",
    "    # Define Q matrix\n",
    "    Q = np.array([[np.cos(theta), 0],\n",
    "                  [0, np.sin(theta)]])\n",
    "    # Define R matrix\n",
    "    R = np.array([[1]])\n",
    "    # Calculate J (for simplicity, assuming a constant value)\n",
    "    J = np.sum(Q) + np.sum(R)\n",
    "    J_values.append(J)\n",
    "\n",
    "# Plotting\n",
    "plt.plot(theta_range, J_values)\n",
    "plt.xlabel('Theta')\n",
    "plt.ylabel('Performance Index (J)')\n",
    "plt.title('Performance Index vs. Parameter Theta')\n",
    "plt.grid(True)\n",
    "plt.show()\n"
   ]
  }
 ],
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