{ "cells": [ { "cell_type": "markdown", "metadata": { "id": "9irECTNCWOVp", "slideshow": { "slide_type": "slide" } }, "source": [ "# Modelizado Experimental\n", "\n", "Vamos a encontrar el modelo matemático del siguiente sistema usando datos experimentales. \n", "\n", "\n", "\n", "La planta es un sistema de iluminación con un LED de potencia (actuador) y un LDR (sensor). Este sistema es controlado a través de un Arduino Mega (actualmente Arduino Mega, en la foto Arduino Uno).\n", "\n", "Un versión para MATLAB puede ser consultada [aquí](http://cpm222.davinsony.com/modelizado/)" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "id": "8e8LHF-IUJ-y", "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "%%capture --no-display\n", "\n", "import warnings\n", "warnings.filterwarnings(action='ignore')\n", "\n", "try: \n", " from control.matlab import *\n", "except:\n", " !pip install control\n", " from control.matlab import *\n", "\n", "import matplotlib.pyplot as plt\n", "plt.rcParams[\"figure.figsize\"] = (4,3)\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import pandas as pd\n", "import scipy.optimize as op \n", "\n", "sp.init_printing()" ] }, { "cell_type": "markdown", "metadata": { "id": "chPsoakTWw5b", "slideshow": { "slide_type": "slide" } }, "source": [ "## Toma de datos\n", "\n", "Para la toma de datos usaremos el código en [lazo_abierto](lazo_abierto/lazo_abierto.ino) para Arduino. Y seguiremos los pasos mostrado a continuación.\n", "\n", "- Descargamos el código.\n", "- Subimos el código al Arduino.\n", "- Abrimos el monitor serial.\n", "- Copiamos los datos de un ciclo completo de apagado y prendido.\n", "- Pegamos en un archivo llamado `datos.csv` con nombre de las columnas `t,u,y`.\n", "- Finalmente, importamos los datos en el Colaboratory o Jupyter Notebook con pandas. " ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "id": "a28YPO2YVjyK", "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "datos = pd.read_csv(\"datos.csv\")" ] }, { "cell_type": "markdown", "metadata": { "id": "nBjrOyZpYDkS", "slideshow": { "slide_type": "subslide" } }, "source": [ "A continuación se muestran algunos datos. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 424 }, "id": "xCf92yrHWJ35", "outputId": "458039ec-3d48-4e62-cd7a-0655f0310d20", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " t u y\n", "0 60968004 0 455\n", "1 60972004 0 454\n", "2 60976004 0 455\n", "3 60980004 0 455\n", "4 60984004 0 455\n", ".. ... ... ...\n", "253 61980004 255 653\n", "254 61984004 255 653\n", "255 61988004 255 653\n", "256 61992004 255 653\n", "257 61996004 255 653\n", "\n", "[258 rows x 3 columns]" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "datos" ] }, { "cell_type": "markdown", "metadata": { "id": "bfCc-ZN2YMt4", "slideshow": { "slide_type": "slide" } }, "source": [ "## Modelizado experimental\n", "\n", "Despues de importar los datos, los extraemos en vectores y graficamos." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 448 }, "id": "Cm5CfsLDWKZL", "outputId": "f5ef8927-ccc0-4136-eb8c-42c84c636117", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "traw = np.array(datos[\"t\"].tolist())\n", "uraw = np.array(datos[\"u\"].tolist())\n", "yraw = np.array(datos[\"y\"].tolist())\n", "\n", "plt.plot(traw,uraw,traw,yraw);" ] }, { "cell_type": "markdown", "metadata": { "id": "IxC-BywsYZ-3", "slideshow": { "slide_type": "subslide" } }, "source": [ "Para la construcción del modelo necesitamos tratar los datos, cambiaremos las unidades del tiempo que esta en microsegundos. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 430 }, "id": "k_dgD3o8YVDD", "outputId": "5375e879-11f2-4555-ca73-13659dbe1359", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t = (traw - traw[0])/1e6\n", "u = (uraw - uraw[0])\n", "y = (yraw - yraw[0])\n", "\n", "plt.plot(t,u,t,y);\n", "plt.xlim((0,0.2));" ] }, { "cell_type": "markdown", "metadata": { "id": "LjezKflQZIZx", "slideshow": { "slide_type": "subslide" } }, "source": [ "Por la respuesta del sistema podemos suponer que es un sistema de primer orden con retardo. La función de transferencia para dicho sistema está dada por la expresión: \n", "\n", "$$G(s) = \\frac{\\gamma}{\\tau\\,s+1}e^{-\\theta\\,s}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "Podemos suponer los valores de los parámetros y modificarlos hasta que las gráficas se parezcan. " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 40 }, "id": "j5rUWfoCY3E3", "outputId": "16cf14d6-ba58-4c6f-cc37-9c6bf91b291d", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "------------------------\n", "Función de transferencia\n", "\n" ] }, { "data": { "text/latex": [ "$$\\frac{0.7765}{0.004438 s + 1}$$" ], "text/plain": [ "TransferFunction(array([0.77647059]), array([0.0044376, 1. ]))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "s = tf(\"s\")\n", "\n", "gamma = y[-1]/u[-1]\n", "tau = 0.0044376\n", "theta = 0.004\n", "\n", "num,den = pade(theta,n=5)\n", "retardo = tf(num,den)\n", "\n", "G = gamma/(tau*s+1)\n", "\n", "print(\"------------------------\")\n", "print(\"Función de transferencia\\n\")\n", "display(G)\n", "GR = G*retardo" ] }, { "cell_type": "markdown", "metadata": { "id": "xCd357ifcLlu", "slideshow": { "slide_type": "subslide" } }, "source": [ "La anterior expresión toma valores encontrados al tanteo. " ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 430 }, "id": "CTxdlBS2cIsD", "outputId": "c5a83275-95f3-46de-e5b3-cd71c5495eca", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "T = np.linspace(t[0],t[-1],num=len(t))\n", "ysim,tsim,_ = lsim(GR,U=u,T=T)\n", "\n", "plt.plot(t,y,tsim,ysim,t,u);\n", "plt.xlim((0,0.2));" ] }, { "cell_type": "markdown", "metadata": { "id": "z-a2DTdfc6Ac", "slideshow": { "slide_type": "subslide" } }, "source": [ "Se ve que siguen habiendo diferencias entre el modelo y los datos. Aquí, podemos contruir un algoritmo de optimización que minimize el error cuadratico entre las dos respuestas (modelo y datos). \n", "\n", "Estos son los resultados:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "xKc4awZvcXMn", "outputId": "a83236e8-35fc-410d-b65d-74e7f92ceafe", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/plain": [ " fun: 171.07466223467733\n", " hess_inv: array([[ 3.05984796e-08, 1.03288890e-09, -3.17140679e-10],\n", " [ 1.03288889e-09, 3.30702396e-09, -2.06624838e-09],\n", " [-3.17140679e-10, -2.06624838e-09, 1.95079611e-09]])\n", " jac: array([-0.18247032, 0.91336441, 2.62592506])\n", " message: 'Desired error not necessarily achieved due to precision loss.'\n", " nfev: 127\n", " nit: 8\n", " njev: 27\n", " status: 2\n", " success: False\n", " x: array([0.77663225, 0.00502317, 0.00301336])" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def model(x):\n", " gamma, tau, theta = x\n", " theta = max(abs(theta),0.001)\n", " G = gamma/(tau*s+1)\n", " num,den = pade(theta,n=5)\n", " retardo = tf(num,den)\n", " GR = G*retardo\n", " ysim,tsim,_ = lsim(GR,U=u,T=T)\n", " return np.sum((y-ysim)**2)\n", "\n", "result = op.minimize(model,[0.77,0.005,0.004])\n", "result" ] }, { "cell_type": "markdown", "metadata": { "id": "0V1RN5MMfcgl", "slideshow": { "slide_type": "subslide" } }, "source": [ "En el anterior resultado encontramos los valores optimizados para los parametros del modelo $\\gamma$, $\\tau$ y $\\theta$. Con estos parámetros podemos verificar la respuesta nuevamente." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "WRpNMTPUdQAJ", "outputId": "e781edfa-de0b-4e96-92a5-05253df10556", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Retardo: 0.0030133589411103744\n" ] } ], "source": [ "gamma, tau, theta = result.x\n", "num,den = pade(theta,n=5)\n", "retardo = tf(num,den)\n", "\n", "G = gamma/(tau*s+1)\n", "GR = G*retardo\n", "\n", "print(\"Retardo:\",theta)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 430 }, "id": "lKqi60duejGD", "outputId": "761ff1e7-e434-4f11-d7ef-275808b2c17c", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "T = np.linspace(t[0],t[-1],num=len(t))\n", "ysim,tsim,_ = lsim(GR,U=u,T=T)\n", "\n", "plt.plot(t,y,tsim,ysim,t,u);\n", "plt.xlim((0,0.2));" ] }, { "cell_type": "markdown", "metadata": { "id": "dOWblJFpfwmt", "slideshow": { "slide_type": "slide" } }, "source": [ "## Diseño de un controlador (PID)\n", "\n", "Para el diseño del controlador podemos usar un controlador PID o diseñar un controlador usando la representación en espacio de estados. Primero usaremos un PID, en este caso usamos un controlador PI, y los parámetros fueron ajustados manualmente. \n", "\n", "[código arduino lazo cerrado](lazo_cerrado/lazo_cerrado.ino)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 452 }, "id": "L--Xn37CevJs", "outputId": "c123e294-f69b-4a9e-dc64-9d93fb2ffe1c", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Controlador PI :\n" ] }, { "data": { "text/latex": [ "$$\\frac{2.5 s + 10}{s}$$" ], "text/plain": [ "TransferFunction(array([ 2.5, 10. ]), array([1., 0.]))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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AeXk5UlNTAQA2my1ozvQ999yDlpYWrF27Fr/61a8wYsQI3HLLLXjqqaf67yh6EBjMhxscGG0y9mk/l9uDh179GIdtDiQOM+BPBdfDHN+3fYmI+pNGDOaYQ5gcDgdMJhPsdjvi4+P7vN+6HUfxn+996X/99f+b26dFj377zufYVPkNjLFabPl5FqaljAin2UREPeprroU1u0NWLec6Llhn095qbKr8BgDwh/nXMqCJKKIuqZDucPf+j4adR77Dqr8eAgCsyJuKvKu7zzwhIhpMqg7priM5HW5Pj3VrT57B0tes8AhgfuZYLLlxwkA3j4joglQd0l25euhJn213Y8krVbCf9c6F/o87M/h7hEQUFVQd0l0viXZ4uvekhRBY+fanOGRzIGGoHqU/mQ5DjG6QWkhE1DtVh3RXoXrSf9p/HNs+roNWA/xxwXVc0Y6IooqqQ7prJHcdk/6szo7/+OthAN4LhdkTEwepZUREfaPukO6S0i7P+YI2pwsPvWZFu9uDW9PN+FkOLxQSUfRRdUh3FdiT/r/lh1Hd5P3pq//84TW8UEhEUUnVId115bumFicAYNeR7/DqB95b13//o2lck4OIopaqQ7qr+zd/jHc/s+GBzR8DAO7JHo/sSRyHJqLopep1N0OtSlL4J29AX5Ucj998f+ogt4iISJlLqicdaNW/ZmCInvOhiSi6qTqke1qpI9lkxPRxIwazKUREYVF1SPfk9Z9ncTYHEUlB3SHdw1LZYy7jXYVEJAd1h3QIc6aO6tPC/0RE0UDVIR2qH73uJ9MHvR1EROFSd0iHSGljLGd0EJE8VB3SRESyU3VId70tnIhINqoOaSIi2ak6pHuYgUdEJA1VhzQRkexUHdJdO9I/nTUuIu0gIgqXqkO6q6LbpkS6CUREiqg6pAPHpOP0Oozk4v5EJBl1h3TAgAdvBCciGak6pANx1TsiklFYIb1+/XqkpaXBaDTCYrFg9+7dvdZ3Op1YuXIlUlNTYTAYMHHiRGzcuDGsBisSMNzBjCYiGSn++awtW7Zg2bJlWL9+PW644QY8//zzyMvLw6FDhzBuXOjZE/Pnz8eJEydQVlaGSZMmobGxES6X66IbrwQzmohkpDikV69ejYKCAixevBgAsGbNGrz33nsoLS1FSUlJt/rvvvsudu7ciWPHjmHkyJEAgPHjx19cq/socAoehzuISEaKhjva29tRVVWF3NzcoPLc3FxUVlaG3Oedd95BZmYmnn76aYwZMwZXXHEFHn74YZw9e7bH73E6nXA4HEHbxWJGE5GMFPWkm5qa4Ha7YTabg8rNZjMaGhpC7nPs2DHs2bMHRqMRb731FpqamvCLX/wCJ0+e7HFcuqSkBE888YSSpoUkBGd3EJHcwrpw2HXoQAjR43CCx+OBRqPB5s2bMXPmTMydOxerV6/Gpk2beuxNFxcXw263+7fa2tpwmtlrm4mIZKCoJ52YmAidTtet19zY2Nitd+2TlJSEMWPGwGQy+cvS09MhhMC3336LyZMnd9vHYDDAYDAoaVpIgTezMKKJSEaKetJ6vR4WiwUVFRVB5RUVFcjOzg65zw033ID6+nq0trb6y44cOQKtVouxY8eG0eS+44VDIpKd4uGOoqIivPjii9i4cSMOHz6M5cuXo6amBoWFhQC8QxULFy7011+wYAESEhJw77334tChQ9i1axceeeQR3HfffRgyZPB+tZsZTUQyUjwFLz8/H83NzVi1ahVsNhsyMjJQXl6O1NRUAIDNZkNNTY2//rBhw1BRUYGHHnoImZmZSEhIwPz58/Hkk0/231H0IHC4gz8QTkQy0ggR/UvjOxwOmEwm2O12xMfH93m/VX85hI17qwEAo+ON2P/onIFqIhGRIn3NNVWv3RG0wBJ70kQkIVWHdCBmNBHJSNUhHTQFj11pIpKQqkM6EDOaiGTEkCYiimKqDunAiStapjQRSUjVIR2IEU1EMlJ1SPO2cCKSnapDOhAjmohkpOqQDrqXkilNRBJSdUgH4oVDIpKRqkM66LbwCLaDiChcqg7pQLG6S+ZQiUhFVJ1cgWPSJT+4OnINISIKk7pDuvNx6S2TMC1lRCSbQkQUFlWHtE8MhzqISFKqTq/o/zkDIqLeqTqkfTizg4hkpfKQZleaiOSm8pD24n0sRCQrVYc0x6SJSHaqDmkfroBHRLJSdUizJ01EslN3SPPCIRFJTtUh7cPRDiKSlapDmsMdRCQ7VYe0j4a3sxCRpFQd0uxIE5HsVB3SPhyTJiJZhRXS69evR1paGoxGIywWC3bv3t2n/fbu3YuYmBhce+214XytYhyTJiLZKQ7pLVu2YNmyZVi5ciWsVitycnKQl5eHmpqaXvez2+1YuHAh5syZE3Zjw8WONBHJSnFIr169GgUFBVi8eDHS09OxZs0apKSkoLS0tNf9lixZggULFiArKyvsxirFedJEJDtFId3e3o6qqirk5uYGlefm5qKysrLH/V566SV8/fXXePzxx/v0PU6nEw6HI2gLS2dGc0yaiGSlKKSbmprgdrthNpuDys1mMxoaGkLu89VXX2HFihXYvHkzYmJi+vQ9JSUlMJlM/i0lJUVJM7vhFDwiklVYFw67LlgkhAi5iJHb7caCBQvwxBNP4Iorrujz5xcXF8Nut/u32tracJrJwQ4ikl7furadEhMTodPpuvWaGxsbu/WuAaClpQUHDhyA1WrFgw8+CADweDwQQiAmJgbbt2/HLbfc0m0/g8EAg8GgpGm94nAHEclKUU9ar9fDYrGgoqIiqLyiogLZ2dnd6sfHx+PTTz/FwYMH/VthYSGmTJmCgwcP4vrrr7+41l+A4Bw8IpKcop40ABQVFeHuu+9GZmYmsrKy8MILL6CmpgaFhYUAvEMVdXV1ePnll6HVapGRkRG0/6hRo2A0GruVExFRd4pDOj8/H83NzVi1ahVsNhsyMjJQXl6O1NRUAIDNZrvgnOnBwn40EclOIyQYE3A4HDCZTLDb7YiPj+/zfr983Yo/H6zHv//LlSiYnTaALSQiUqavuabqtTui/68fIqLeqTukOx85uYOIZKXqkPbhFDwikpWqQ1qC4XYiol6pOqR92JEmIlmpOqTZjyYi2ak6pH1CrStCRCQDdYc0u9JEJDl1h3QndqSJSFaqDmn+MgsRyU7dIe37ZZbINoOIKGyqDmk/jncQkaRUHdK8l4WIZKfqkPZhP5qIZKXqkOaFQyKSnapD2odD0kQkK1WHNMekiUh2qg5pHw1HpYlIUqoOaXakiUh26g5p380s7EgTkaRUHdI+zGgikpXKQ5oDHkQkN5WHtBeHO4hIVqoOaU7BIyLZqTqkfTgFj4hkpeqQZkeaiGSn6pD2Y0eaiCSl6pAWHJQmIsmpO6Q7H9mRJiJZqTqkfTScg0dEkgorpNevX4+0tDQYjUZYLBbs3r27x7rbtm3Dbbfdhssvvxzx8fHIysrCe++9F3aDleBoBxHJTnFIb9myBcuWLcPKlSthtVqRk5ODvLw81NTUhKy/a9cu3HbbbSgvL0dVVRVuvvlmzJs3D1ar9aIb31fsRxORrDRC4dW166+/HtOnT0dpaam/LD09HXfeeSdKSkr69BlXXXUV8vPz8dhjj4V83+l0wul0+l87HA6kpKTAbrcjPj6+z21duPFD7DryHX7/o2m4yzK2z/sREQ00h8MBk8l0wVxT1JNub29HVVUVcnNzg8pzc3NRWVnZp8/weDxoaWnByJEje6xTUlICk8nk31JSUpQ008/39w+HpIlIVopCuqmpCW63G2azOajcbDajoaGhT5/x+9//Hm1tbZg/f36PdYqLi2G32/1bbW2tkmZ2w5AmIlnFhLNT19kSQog+zaB47bXX8Nvf/hZ//vOfMWrUqB7rGQwGGAyGcJpGRKQqikI6MTEROp2uW6+5sbGxW++6qy1btqCgoABvvPEGbr31VuUtvQhcu4OIZKVouEOv18NisaCioiKovKKiAtnZ2T3u99prr+Gee+7Bq6++ijvuuCO8loaBU/CISHaKhzuKiopw9913IzMzE1lZWXjhhRdQU1ODwsJCAN7x5Lq6Orz88ssAvAG9cOFCPPvss5g1a5a/Fz5kyBCYTKZ+PJTuBHjhkIjkpjik8/Pz0dzcjFWrVsFmsyEjIwPl5eVITU0FANhstqA5088//zxcLhceeOABPPDAA/7yRYsWYdOmTRd/BEREKqZ4nnQk9HU+YVcL/ms/Kr9uxrM/vhb/eu2YAWwhEZEyAzJPWlZcu4OIZKXqkI7+fyMQEfVO1SHtw340EclK1SEt+ANaRCQ5dYd0Z0ZzSJqIZKXqkPbhHYdEJCtVhzQHO4hIdqoOaR8OdxCRrNQd0uxKE5HkVB3S/rU7ItwOIqJwqTqkfTjcQUSyUnVI845DIpKdqkP6PHaliUhOqg5pdqSJSHbqDmn+WjgRSU7VIe3DjCYiWak6pDncQUSyU3VI+3DRfyKSlapDmlPwiEh26g7pzkf2o4lIVqoOaR+OdhCRrNQd0hzvICLJqTukO7EnTUSyUnVIsx9NRLJTd0j7fuOQlw6JSFKqDmk/ZjQRSUrVIS044EFEklN1SJ9q6wDAjjQRyUvVIV13+iwAoNHhjHBLiIjCE1ZIr1+/HmlpaTAajbBYLNi9e3ev9Xfu3AmLxQKj0YgJEyZgw4YNYTU2XJ/X2wf1+4iI+ovikN6yZQuWLVuGlStXwmq1IicnB3l5eaipqQlZv7q6GnPnzkVOTg6sViseffRRLF26FFu3br3oxvfG4zk/Hs2RaSKSleKQXr16NQoKCrB48WKkp6djzZo1SElJQWlpacj6GzZswLhx47BmzRqkp6dj8eLFuO+++/DMM89cdON7c6LlnP956znXgH4XEdFAURTS7e3tqKqqQm5ublB5bm4uKisrQ+6zb9++bvVvv/12HDhwAB0dHSH3cTqdcDgcQZtSLndA/5lXDolIUopCuqmpCW63G2azOajcbDajoaEh5D4NDQ0h67tcLjQ1NYXcp6SkBCaTyb+lpKQoaSYAIEZ3PpmDApuISCJhXTjsuoi+EKLXhfVD1Q9V7lNcXAy73e7famtrFbfRPNzof+72MKSJSE4xSionJiZCp9N16zU3NjZ26y37jB49OmT9mJgYJCQkhNzHYDDAYDAoaVo3Wu35vwCG6HUX9VlERJGiqCet1+thsVhQUVERVF5RUYHs7OyQ+2RlZXWrv337dmRmZiI2NlZhc8MzdfTwQfkeIqL+pni4o6ioCC+++CI2btyIw4cPY/ny5aipqUFhYSEA71DFwoUL/fULCwtx/PhxFBUV4fDhw9i4cSPKysrw8MMP999R9ODpH16D3CvN+Oms1AH/LiKigaBouAMA8vPz0dzcjFWrVsFmsyEjIwPl5eVITfUGoc1mC5oznZaWhvLycixfvhzr1q1DcnIynnvuOdx11139dxQ9mJ+ZgvmZyi86EhFFC40Q0f/zJQ6HAyaTCXa7HfHx8ZFuDhHRRetrrql67Q4iItkxpImIohhDmogoijGkiYiiGEOaiCiKMaSJiKKY4nnSkeCbJRjOanhERNHIl2cXmgUtRUi3tLQAQFir4RERRbOWlhaYTKYe35fiZhaPx4P6+noMHz6819X2unI4HEhJSUFtba2qboLhcclHrcfG4wqfEAItLS1ITk6GVtvzyLMUPWmtVouxY8eGvX98fLyq/gfy4XHJR63HxuMKT289aB9eOCQiimIMaSKiKKbqkDYYDHj88ccv+gcEog2PSz5qPTYe18CT4sIhEdGlStU9aSIi2TGkiYiiGEOaiCiKMaSJiKIYQ5qIKIpJH9Lr169HWloajEYjLBYLdu/e3Wv9nTt3wmKxwGg0YsKECdiwYcMgtVQZJcf1/vvvQ6PRdNu++OKLQWzxhe3atQvz5s1DcnIyNBoN3n777QvuI8P5UnpcspyvkpISzJgxA8OHD8eoUaNw55134ssvv7zgftF+zsI5rkieM6lDesuWLVi2bBlWrlwJq9WKnJwc5OXlBf1aeaDq6mrMnTsXOTk5sFqtePTRR7F06VJs3bp1kFveO6XH5fPll1/CZrP5t8mTJw9Si/umra0N06ZNw9q1a/tUX5bzpfS4fKL9fO3cuRMPPPAA9u/fj4qKCrhcLuTm5qKtra3HfWQ4Z+Ecl09EzpmQ2MyZM0VhYWFQ2dSpU8WKFStC1v/1r38tpk6dGlS2ZMkSMWvWrAFrYziUHteOHTsEAHHq1KlBaF3/ACDeeuutXuvIcr4C9eW4ZDxfQgjR2NgoAIidO3f2WEfGc9aX44rkOZO2J93e3o6qqirk5uYGlefm5qKysjLkPvv27etW//bbb8eBAwfQ0dExYG1VIpzj8rnuuuuQlJSEOXPmYMeOHQPZzEEhw/m6GLKdL7vdDgAYOXJkj3VkPGd9OS6fSJwzaUO6qakJbrcbZrM5qNxsNqOhoSHkPg0NDSHru1wuNDU1DVhblQjnuJKSkvDCCy9g69at2LZtG6ZMmYI5c+Zg165dg9HkASPD+QqHjOdLCIGioiLMnj0bGRkZPdaT7Zz19bgiec6kWKq0N13XlxZC9LrmdKj6ocojTclxTZkyBVOmTPG/zsrKQm1tLZ555hnceOONA9rOgSbL+VJCxvP14IMP4pNPPsGePXsuWFemc9bX44rkOZO2J52YmAidTtetd9nY2Njtb3Kf0aNHh6wfExODhISEAWurEuEcVyizZs3CV1991d/NG1QynK/+Es3n66GHHsI777yDHTt2XHBdd5nOmZLjCmWwzpm0Ia3X62GxWFBRURFUXlFRgezs7JD7ZGVldau/fft2ZGZmIjY2dsDaqkQ4xxWK1WpFUlJSfzdvUMlwvvpLNJ4vIQQefPBBbNu2Df/4xz+QlpZ2wX1kOGfhHFcog3bOBv1SZT96/fXXRWxsrCgrKxOHDh0Sy5YtE0OHDhXffPONEEKIFStWiLvvvttf/9ixYyIuLk4sX75cHDp0SJSVlYnY2Fjx5ptvRuoQQlJ6XH/4wx/EW2+9JY4cOSI+++wzsWLFCgFAbN26NVKHEFJLS4uwWq3CarUKAGL16tXCarWK48ePCyHkPV9Kj0uW83X//fcLk8kk3n//fWGz2fzbmTNn/HVkPGfhHFckz5nUIS2EEOvWrROpqalCr9eL6dOnB02jWbRokbjpppuC6r///vviuuuuE3q9XowfP16UlpYOcov7RslxPfXUU2LixInCaDSKyy67TMyePVv87W9/i0Cre+ebxtR1W7RokRBC3vOl9LhkOV+hjgmAeOmll/x1ZDxn4RxXJM8Z15MmIopi0o5JExFdChjSRERRjCFNRBTFGNJERFGMIU1EFMUY0kREUYwhTUQUxRjSRERRjCFNRBTFGNJERFGMIU1EFMX+F5mS8T2ZYE6jAAAAAElFTkSuQmCC", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Kp = 2.5\n", "Ki = 10\n", "Kd = 0.00\n", "K = Kp + Ki/s + Kd*s\n", "\n", "print(\"Controlador PI :\")\n", "display(K)\n", "CPID = feedback(K*GR,1)\n", "yy,tt = step(CPID)\n", "plt.plot(tt,yy); " ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "q4YX_G6cmg43", "outputId": "173a70fb-c265-4666-c739-3e459805d6d3", "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Información de la respuesta a un escalón:\n", "\n" ] }, { "data": { "text/plain": [ "{'RiseTime': 0.0025980221962962294,\n", " 'SettlingTime': 1.0631106827244172,\n", " 'SettlingMin': 0.4995665080555242,\n", " 'SettlingMax': 1.0219807319464205,\n", " 'Overshoot': 2.1980731946420473,\n", " 'Undershoot': 2.024617961818068,\n", " 'Peak': 1.0219807319464205,\n", " 'PeakTime': 0.007274462149629443,\n", " 'SteadyStateValue': 1.0}" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "'''\n", "plt.axhline(y=0, color='k', linestyle='-')\n", "plt.axvline(x=0, color='k', linestyle='-')\n", "plt.plot(np.real(zero(retardo)),np.imag(zero(retardo)),\"ob\");\n", "plt.plot(np.real(pole(retardo)),np.imag(pole(retardo)),\"xb\");\n", "plt.plot(np.real(zero(G)),np.imag(zero(G)),\"or\");\n", "plt.plot(np.real(pole(G)),np.imag(pole(G)),\"xr\");\n", "plt.plot(np.real(zero(CPID)),np.imag(zero(CPID)),\"og\");\n", "plt.plot(np.real(pole(CPID)),np.imag(pole(CPID)),\"xg\");\n", "plt.xlim((-4000,0))\n", "'''\n", "print(\"Información de la respuesta a un escalón:\\n\")\n", "\n", "stepinfo(CPID)" ] }, { "cell_type": "markdown", "metadata": { "id": "5sO-arGOup3O", "slideshow": { "slide_type": "slide" } }, "source": [ "# Diseño de un controlador por Espacio de Estados\n", "\n", "Para el diseño del controlador, debemos convertir la función de transferencia en una representación en espacio de estados:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 58 }, "id": "gq2MoFAihknP", "outputId": "7307c8ac-4eb5-4d78-d9c2-b8eec5469a61", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$$\n", "\\left(\\begin{array}{rll|rll}\n", "-199\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\hline\n", "155\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\end{array}\\right)\n", "$$" ], "text/plain": [ "StateSpace(array([[-199.07744763]]), array([[1.]]), array([[154.60996579]]), array([[0.]]))" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SS = tf2ss(G)\n", "SS" ] }, { "cell_type": "markdown", "metadata": { "id": "UEWQ_4IPu38q", "slideshow": { "slide_type": "subslide" } }, "source": [ "Donde la Matriz A es: " ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 39 }, "id": "XWCbg-jHs7Wa", "outputId": "79b654e3-f60c-41da-9859-9c0ce2e5d0a6", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}-199.077447630764\\end{matrix}\\right]$" ], "text/plain": [ "[-199.077447630764]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "A = sp.Matrix(SS.A)\n", "B = sp.Matrix(SS.B)\n", "C = sp.Matrix(SS.C)\n", "\n", "display(A)" ] }, { "cell_type": "markdown", "metadata": { "id": "JNgvFa-qu7sC", "slideshow": { "slide_type": "-" } }, "source": [ "Aquí tenemos que el valor propio de la matriz es igual al polo del sistema representado en función de transferencia." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "ebmX2dwpvFNb", "outputId": "534da0ad-aefd-4174-9ad9-8449592823fd", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/plain": [ "array([-199.07744763+0.j])" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pole(G)" ] }, { "cell_type": "markdown", "metadata": { "id": "eLL8cN3nvIiQ", "slideshow": { "slide_type": "subslide" } }, "source": [ "Para el diseño de un controlador en espacio de estados podemos verificar si el sistema es completamente controlable, calculando la matriz de controlabilidad y verificando que el rango de la matriz es igual al número de variables de estado. En este caso el rango es $1$." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "3lshcU93vPxm", "outputId": "c32874c9-8f0a-4da6-facb-fb0273f36f51", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/plain": [ "matrix([[1.]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Gamma = ctrb(SS.A,SS.B)\n", "display(Gamma)" ] }, { "cell_type": "markdown", "metadata": { "id": "qu_z20JSvpQ8", "slideshow": { "slide_type": "subslide" } }, "source": [ "Teniendo un sistema completamente controlable, podemos ubicar los polos del sistema en lazo cerrado donde queramos. Recordemos que para esto la realimentación se hace desde el vector de estado:\n", "\n", "![Lazo Cerrado](lazo-cerrado.png)\n", "\n", "Para esto vamos a calcular la ecuación característica del sistema en lazo cerrado con variable en los elementos del controlador y vamos a igualarlas con la ecuación caracteristica resultante del posicionamiento de polos. \n", "\n", "Para calcular la ecuación caracteristica usaremos la expresión\n", "\n", "$$\\text{det}\\left(s\\mathbb{I}-(A-BK)\\right) = 0$$" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 273 }, "id": "t_kHZpdftoAT", "outputId": "a4a7933a-4dbb-4ff1-aaf6-7ae516ac4ef7", "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "--------\n", "Matriz K\n", "\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k\\end{matrix}\\right]$" ], "text/plain": [ "[k]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "-----------\n", "Matriz A-BK\n", "\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}- 1.0 k - 199.077447630764\\end{matrix}\\right]$" ], "text/plain": [ "[-k - 199.077447630764]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "-----------------------\n", "Ecuación característica\n", "\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle 1.0 k + s + 199.077447630764 = 0$" ], "text/plain": [ "1.0⋅k + s + 199.077447630764 = 0" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "k,symS = sp.symbols(\"k,s\")\n", "K = sp.Matrix([[k]])\n", "\n", "print(\"--------\")\n", "print(\"Matriz K\\n\")\n", "display(K)\n", "\n", "Ac = A-B*K\n", "\n", "print(\"\\n-----------\")\n", "print( \"Matriz A-BK\\n\")\n", "display(Ac)\n", "\n", "det = sp.Eq((symS*sp.eye(1) - Ac).det(),0)\n", "\n", "print(\"\\n-----------------------\")\n", "print( \"Ecuación característica\\n\")\n", "display(det)" ] }, { "cell_type": "markdown", "metadata": { "id": "2CjwDXZQxzZu", "slideshow": { "slide_type": "subslide" } }, "source": [ "Si quisieramos ubicar el polo en $s=-250$ deberemos tener un polinomio tipo:\n", "\n", "$$s+250 = 0$$\n", "\n", "luego:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 38 }, "id": "x_5dntscxqc3", "outputId": "c8f97589-197d-4516-f5a2-3161472b9ec1", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle 1.0 k + 199.077447630764 = 250$" ], "text/plain": [ "1.0⋅k + 199.077447630764 = 250" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "comp1 = sp.Eq(sp.Poly(det.lhs,symS).coeffs()[1],250)\n", "display(comp1)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "de aquí:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 38 }, "id": "iK4-k-lGyHTP", "outputId": "8cbd333f-56e8-44fa-aa50-56d09b67dcd2", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle k = 50.9225523692361$" ], "text/plain": [ "k = 50.9225523692361" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sol = sp.Eq(k,sp.solve(comp1,k)[0])\n", "display(sol)" ] }, { "cell_type": "markdown", "metadata": { "id": "eHxtJTN_zKYG", "slideshow": { "slide_type": "subslide" } }, "source": [ "Remplazando el valor de $k$ en la matriz dinámica en lazo cerrado tendremos: " ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 39 }, "id": "CWovLfnNybtr", "outputId": "bcd32f99-6706-4bc4-d3fd-8dfdc8045fe6", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}-250.0\\end{matrix}\\right]$" ], "text/plain": [ "[-250.0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Ac = Ac.subs({k:sol.rhs})\n", "display(Ac)" ] }, { "cell_type": "markdown", "metadata": { "id": "YxAxDeo0zlTY", "slideshow": { "slide_type": "-" } }, "source": [ "Tendremos entonces una representación en espacio de estados para el sistema en lazo cerrado de la siguiente forma: " ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 58 }, "id": "gN8Xb1gZzCF5", "outputId": "15238790-5262-440c-941a-74314e865257", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$$\n", "\\left(\\begin{array}{rll|rll}\n", "-250\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\hline\n", "155\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\end{array}\\right)\n", "$$" ], "text/plain": [ "StateSpace(array([[-250.]]), array([[1.]]), array([[154.60996579]]), array([[0.]]))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "SS.A = np.array(Ac).astype(np.float64)\n", "display(SS)" ] }, { "cell_type": "markdown", "metadata": { "id": "0l_ufxcmzwRF", "slideshow": { "slide_type": "subslide" } }, "source": [ "Con este tipo de sistema, la constante de tiempo del sistema es: \n", "\n", "$$\\tau = 1/250 = 4 \\text{ milisegundos}$$\n", "\n", "Es decir un tiempo de establecimiento de $t_s = 20 \\text{ ms}$" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 430 }, "id": "uMFWNKCbz5u5", "outputId": "82c5184f-b4ab-439f-e718-fd063a36a385", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "y,t = step(SS)\n", "plt.plot(t,y);" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Podemos ver que el tiempo de respuesta es correcto pero el sistema no llega al objetivo. " ] }, { "cell_type": "markdown", "metadata": { "id": "0A4qeN990Pod", "slideshow": { "slide_type": "subslide" } }, "source": [ "Para esto tenedremos que generar otro lazo cerrado e integrar la señal de error $e$. \n", "\n", "![Lazo Cerrado con Integrador](lazo-cerrado-con-integrador.png)" ] }, { "cell_type": "markdown", "metadata": { "id": "_RrXD3tA1rRQ", "slideshow": { "slide_type": "subslide" } }, "source": [ "El recuadro azul tiene la siguiente representación en espacio de estados: \n", "\n", "$$\n", "\\begin{align}\n", "\\dot{x} &= (A-BK)\\,x + w \\\\\n", "y &= C\\,x \n", "\\end{align}\n", "$$\n", "\n", "del diagrama de bloques externo al recuadro tendremos las siguientes expresiones:\n", "\n", "$$e=r-y=r-C\\,x=\\dot{x}_N \\qquad w = K_e\\,x_N$$\n", "\n", "combinando las expresiones:\n", "\n", "$$\n", "\\begin{align}\n", "\\dot{x} &= (A-BK)\\,x + K_e\\,x_N \\\\\n", "\\dot{x}_N &= -C\\,x + r \\\\\n", "y &= C\\,x \n", "\\end{align}\n", "$$\n", "\n", "Si representamos el sistema de manera matricial:\n", "\n", "$$\n", "\\begin{align}\n", "\\begin{bmatrix}\n", "\\dot{x} \\\\\n", "\\dot{x}_N\n", "\\end{bmatrix}\n", "&= \n", "\\begin{bmatrix}\n", "A-BK & K_e\\\\\n", "-C & 0\n", "\\end{bmatrix}&\n", "\\begin{bmatrix}\n", "x \\\\\n", "x_N\n", "\\end{bmatrix}&\n", "+\n", "\\begin{bmatrix}\n", "0 \\\\\n", "1\n", "\\end{bmatrix}r \\\\\n", "y &= \\begin{bmatrix}\n", "C & 0\n", "\\end{bmatrix}&\n", "\\begin{bmatrix}\n", "x \\\\\n", "x_N\n", "\\end{bmatrix}&\n", "\\end{align}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "id": "9OCdbzo24e5g", "slideshow": { "slide_type": "subslide" } }, "source": [ "La nueva matriz dinámica para este sistema será:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 58 }, "id": "mxiDSniq0LYJ", "outputId": "c545e942-24c0-4552-c896-0e502a1dd219", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}- 1.0 k - 199.077447630764 & K_{e}\\\\-154.609965793313 & 0\\end{matrix}\\right]$" ], "text/plain": [ "⎡-k - 199.077447630764 Kₑ⎤\n", "⎢ ⎥\n", "⎣ -154.609965793313 0 ⎦" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "ke = sp.symbols(\"K_e\")\n", "\n", "AI = sp.Matrix([[(A-B*K)[0],ke],[-C[0],0]])\n", "display(AI)" ] }, { "cell_type": "markdown", "metadata": { "id": "sqWouFks4jTT", "slideshow": { "slide_type": "-" } }, "source": [ "Con la siguiente ecuación caracteristica:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 38 }, "id": "08RvwQui3_dx", "outputId": "80a94c13-28eb-4f00-8a2e-2934e990936d", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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/FnujHuvuuz22BcYQEWm24WedtJMsIXTgV94tsWlFBNhgwN9+Bdi7luEjW0DbEGRXQ8rFV8chbbYnbVeXy+q115IlzVYQRyKRRqjqaGx10jY5lz8rOZ0SiUQika8p3eXAH4l8nfgcm9p8W3K+txaJRCKRSCPiyFgkUhG1D2DPw/wCJmEfpB2I+fEcIyKxkkUikUikkO7a9DUS+drgViXtjfnUzALmAucBr8eOWCQSiUSaEUfGIpFIJBKJRHqQODIWiUQikUgk0oP8H+Vej4i5oE5XAAAAAElFTkSuQmCC", "text/latex": [ "$\\displaystyle 154.609965793313 K_{e} + 1.0 k s + s^{2} + 199.077447630764 s = 0$" ], "text/plain": [ " 2 \n", "154.609965793313⋅Kₑ + 1.0⋅k⋅s + s + 199.077447630764⋅s = 0" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "det2 = sp.Eq((symS*sp.eye(2)-AI).det(),0)\n", "display(det2)\n", "\n", "P1 = sp.Poly(det2,symS)" ] }, { "cell_type": "markdown", "metadata": { "id": "g9xVLqui6xyw", "slideshow": { "slide_type": "subslide" } }, "source": [ "Creamos una ecuación característica con los polos arbitrarios:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 60 }, "id": "NzMt125s4byk", "outputId": "1a3f855a-0e64-4aef-dc5c-66e1a69ba4b5", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAAUCAYAAAAjk8nJAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAABJ0AAASdAHeZh94AAAGHklEQVR4nO2aeYhVVRzHP6NSipSGlYKRY5oFFVoZKZlloxZtZuI/kbkQJkmTNUKpxM+v0Aahpi1ahloZFQwalJVZU5otpiVlaeaGBaaZ5UZmTdMf545ezzv3LTPznu/hfOBxeGe7v/u9Z/2dU1ZXV0czzTSTmRYn2oBmmikVWoUiJb0M3AB0NbNDhTWpOJB0ObAGuNvMXsqQ96TXK1dy0bdYKPOXYZJ6A6uBiWY2/YRYlQFJ24EuCcm7zKxTQrlzgGm4ht0B2AksAWRmfwTyLwb6AOeb2cGEOotaL0kdgKHATcAlQGfgCPAdMB+Yb2b/BcptJ88aZ6NvIcjW5tAy7DFgP/B8/s1sFPsABX5PhTJL6gasBUbjGvcMYCtwP/B51Kh8Hgc6AZVp7Ch2vYYDLwJXAl8CM4Fq4GJgHvCmpLKEsvnWOBt980ouNrfyCvYABgLzzOyvPBs5CjeyDTCzjxtQxZ9mNjWH/M8BZwOVZjY7Zsd04AHgUWBcvICZrZa0EbhH0pNmVuu9QynotQm4FXgnPoNImoxrHMOA23EdyCevGmfSt0BkbbO/ZxkDlAFvhGqVdHVUQU/cdL4f2A4sN7PJTfoKTYik84DBOFuf9ZINGAuMkFQV2HO8DkzFdYr3vbSi18vMPkqI/1XSHFxjuJZwZ8maRmicTt+8kqvNfmcZCNQCXwQqnowTdgfupfYAHYHeuLVeoTvLqZLuBM4FDgHfAisSRqfronCZvz43swOSVuFE6wN86JVdFYWDSP2YpaRXiH+i8N+E9EJonE7ffJOTzUc7i6S2QC9ggz+6SuqI2wB9ClSY2REv/cymfoss6AS84sVtkzTazD7x4i+Iwk0Jdf2EE6UHqZ3lqyjsH48sQb2OQ1Ir4K7o73sJ2QqhcVDfBJsnAO0z5YuxzsyWpEnPyeb4Br8z0BLnCfC5MErb5H94ADPbk8HopmY+UIH7mG1xXp65QDnwrqSeXv52Ubgvob76+PZ+gpntAw7jRtc4paRXiCdwm/ylZhYa0QuicRp9Q0zALY+y/d2Wob6cbI4vw+p3/SkuVOD7qOAYSWcBi3BTVyhvChnckDWS/LiFZjYqqT4z8wusB8ZJOghU4dbAQ7OxLaLeG5R0nWEvbgkVp2T0CtRfidNpIzAilKfAGof0DdlUnsPzmoLjbI53lnpvTmu/hJntkdQP11tvBG4BaiV9AEwxs68zPHQmqaN2L2AIsBC3wYqzLkN9SczBfUh/Sq8fIdoR5nQvn08bjulTT0nqJWk88DTwA26JuDfbshH50DikbyHIyeZ4Z9kdhaHzBsxsPTBc0ik4ocbifPhXSOpsZn8nWWRmM/24yBU6BFjQQNdxiPp3aOvF/xiFPRLKnR+FKWtXSS1wDXdbwrNKRq9ozT8DN0tUmNnu9CWCNKnGafRNIQ97lpxsjneWncBvHNv0BInW4MuB5ZJWAv1wU+iOtGYXhr5RuNWLr4nCwZJaeOcNpwFX4Ua2FK8WTo8yUkfvktJL0kO4fco6YFAj9k1NrXGSviEmkLw8DbEQdxKfRE42H+0sZlYnaQUwTFJ3M9scFboU2G9mW+JPkdQdt0HcAfySwws0CkkXATv95YOkLsAz0d9X42lmtkXSMpxnYzwwO14UN0rOTbjX1ScKa+KRpaJX9OxHcN65tcDgTEuvAmsc1DdEU+9ZcrXZP2epxp3oXg9sjuIqgZGSVuM2rruBrrhTYYAxobtFeWQ48LCkGtzUfQDohrv71BpYSvg6xr3AZ8AsSRXABtwVkAG4aXZKwvMG485S3gqkFb1ekkbiOkotsBKoDDgItpvZgtj/QmqcTt9CkLXN/t2wamAXx/zv4F5iEXAGTsQq3GjwGtDTzPxziXxTAyzGNcA7gAeBa3BnGiOBmxPctVtwB4ILcGJU4RrALKCvmf3ul5HUDud+fNvMfg7YUgp6dY3CliS7Xkd5ZQqicRb65p1cbA7dOp6Euxx4mZl9UyCbixJJ9+FE629mKxPyNOvVQLLRt5gI3TqegVtXTyuwLUWFpDbAJKA6w4ds1qsB5KBv0ZDSWczsMO6gak10peNkpRx4AZiYLlOzXg2mnCz0LSb+B2HSi5vR5R6GAAAAAElFTkSuQmCC", "text/latex": [ "$\\displaystyle \\left(s + 50\\right) \\left(s + 250\\right) = 0$" ], "text/plain": [ "(s + 50)⋅(s + 250) = 0" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle s^{2} + 300 s + 12500 = 0$" ], "text/plain": [ " 2 \n", "s + 300⋅s + 12500 = 0" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "eq1 = sp.Eq((symS+250)*(symS+50),0)\n", "display(eq1)\n", "eq2 = sp.expand(eq1)\n", "display(eq2)\n", "P2 = sp.Poly(eq2,symS)" ] }, { "cell_type": "markdown", "metadata": { "id": "yVuqjWYS66ro", "slideshow": { "slide_type": "-" } }, "source": [ "Igualamos los coefficientes de las dos ecuaciones características:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 203 }, "id": "HWYFn_C65BXU", "outputId": "aa841fdd-ac91-4839-cb80-12669defc3ac", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "---------------------\n", "Sistema de ecuaciones\n", "\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle 1.0 k + 199.077447630764 = 300$" ], "text/plain": [ "1.0⋅k + 199.077447630764 = 300" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle 154.609965793313 K_{e} = 12500$" ], "text/plain": [ "154.609965793313⋅Kₑ = 12500" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "---------------------------\n", "Valores para el controlador\n", "\n" ] }, { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\left\\{ K_{e} : 80.8486046540516, \\ k : 100.922552369236\\right\\}$" ], "text/plain": [ "{Kₑ: 80.8486046540516, k: 100.922552369236}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cP1 = P1.coeffs()\n", "cP2 = P2.coeffs()\n", "\n", "print(\"---------------------\")\n", "print(\"Sistema de ecuaciones\\n\")\n", "for i,c in enumerate(cP1):\n", " if i > 0:\n", " display(sp.Eq(cP1[i],cP2[i]))\n", " if i == 1:\n", " eqs = {sp.Eq(cP1[i],cP2[i])}\n", " if i > 1:\n", " eqs.add(sp.Eq(cP1[i],cP2[i]))\n", "\n", "print(\"\\n---------------------------\")\n", "print( \"Valores para el controlador\\n\")\n", "\n", "ks = sp.solve(eqs,{k,ke})\n", "display(ks)" ] }, { "cell_type": "markdown", "metadata": { "id": "30uAHPdO7ATs", "slideshow": { "slide_type": "subslide" } }, "source": [ "Encontrando los valores del controlador, remplazamos en la matriz dinámica." ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 92 }, "id": "Sl5_WZBF5181", "outputId": "532bfee0-ce44-4543-ee6d-99c958d7e006", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Matriz A\n", "\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}-300.0 & 80.8486046540516\\\\-154.609965793313 & 0\\end{matrix}\\right]$" ], "text/plain": [ "⎡ -300.0 80.8486046540516⎤\n", "⎢ ⎥\n", "⎣-154.609965793313 0 ⎦" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "AI2 = AI.subs(ks)\n", "AIn = np.array(AI2).astype(np.float64)\n", "print(\"Matriz A\\n\")\n", "display(AI2)" ] }, { "cell_type": "markdown", "metadata": { "id": "yIrnrKZD7Gud", "slideshow": { "slide_type": "-" } }, "source": [ "Verificamos que los valores propios si correspondan a los polos deseados: " ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "AhWYwaPT6U_A", "outputId": "eabcb3b3-6fe9-4b25-f2b2-5139bab237c0", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/plain": [ "(array([-250., -50.]),\n", " array([[-0.8504964 , -0.30770401],\n", " [-0.52598087, -0.95148213]]))" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.linalg.eig(AIn)" ] }, { "cell_type": "markdown", "metadata": { "id": "U4XpUP3s91Ib", "slideshow": { "slide_type": "subslide" } }, "source": [ "Construyamos las matrices adicionales para el sistema:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 92 }, "id": "qc95f4Fm7XqH", "outputId": "96a47d34-9a5b-45a8-d8cf-2bba7d912779", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Matriz B\n", "\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right]$" ], "text/plain": [ "⎡0⎤\n", "⎢ ⎥\n", "⎣1⎦" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BIn = np.matrix([[0],[1]])\n", "print(\"Matriz B\\n\")\n", "sp.Matrix(BIn)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 73 }, "id": "WW-r0VtO76al", "outputId": "4a30fe16-cdca-4ca9-bd8a-499569407228", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Matriz C\n", "\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}154.609965793313 & 0\\end{matrix}\\right]$" ], "text/plain": [ "[154.609965793313 0]" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CIn = np.matrix([[SS.C[0,0],0]])\n", "print(\"Matriz C\\n\")\n", "sp.Matrix(CIn)" ] }, { "cell_type": "markdown", "metadata": { "id": "TRS_jUwI940X", "slideshow": { "slide_type": "subslide" } }, "source": [ "Luego nuestro sistema completo será" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 79 }, "id": "EHs0Bkk58AAP", "outputId": "628a1396-2b10-4f49-9d64-f8c33323c3f7", "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$$\n", "\\left(\\begin{array}{rllrll|rll}\n", "-300\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&80.&\\hspace{-1em}8&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "-155\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\hline\n", "155\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\end{array}\\right)\n", "$$" ], "text/plain": [ "['y[0]']>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "DIn = np.matrix([[0]])\n", "SSI = ss(AIn,BIn,CIn,DIn)\n", "display(SSI)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 447 }, "id": "S3_xAny98d73", "outputId": "210d8fef-7422-43ab-d4f1-3b20ab69c27a", "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Respuesta a un escalón\n" ] }, { "data": { "image/png": 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dw1ytacCyT45BCGDeHVGYxTU5iLwaQ7oHEULgjzv+D2UmMwYP8MXzc0ZKXRIRdRJDugfZnnMJWc3zod+cNw4+al6wQuTtGNI9hMFYhz//Tx4AIC15GOIidBJXRERdgSHdAwgh8Nxnx1FVb8GYqH74bRJXtiPqKRjSPcCO74vxzanLUCvkeO2h26GQcz40UU/BkPZypcZ6vPiPkwCApdOHYmiIn8QVEVFXYkh7uZU7T6Cq3oLbI3X4HYc5iHochrQX+/rHMuw+WQaFXIZXH7odSgXfTqKehp9qL1XXYMXzXzYNcyy+MwYjQnkzBKKeiCHtpdZ8cxaXrtYhTKfFk/cMlbocInIThrQXOltejXf3N63NsXLOSPjyLitEPRZD2ssIIfD8lyfQaBWYOnwAZowKlbokInIjhrSXycwrQ/a5SqiVcrz4c64RTdTTMaS9SIPFhvSMfADAb5NiMDCQS5AS9XQMaS+y+dB5nK+sRVBfDf7jrtukLoeIugFD2ktcrWnAm3vOAACeTh6GvvyykKhXYEh7iTf2nIGp3oIRoX74VTxvKEvUWzCkvcC5y9X4++ELAID/nD2SCygR9SIMaS+wOvM0LDaBu0cE486hQVKXQ0TdiCHt4U4UG/HVcQNkMuCP9w6Xuhwi6mYMaQ+3Ous0AGDO7eFcn4OoF2JIe7CcC1fw9Y/lUMhlWDZ9mNTlEJEEGNIeSgiBv+4+BQD4lT4SMUG+EldERFJwKaTXrl2LmJgYaLVa6PV6HDhw4KbtzWYzVqxYgejoaGg0GgwZMgQbN250qeDe4uDZShwuuAK1Qo4nuModUa/l9BUR27Ztw9KlS7F27VpMnjwZ7777LmbOnIm8vDwMHDiwzX3mzp2LsrIybNiwAbfddhvKy8thsVg6XXxPJYTA6qyms+j5Ewciol8fiSsiIqnIhBDCmR0mTpyI8ePHY926dfZtsbGxuP/++5Gent6q/a5duzBv3jwUFBSgf//+LhVpMpmg0+lgNBrh79/zvzzLPluB+e9/B7VSjm//NBXBflqpSyKiLtbRXHNquKOhoQE5OTlITk522J6cnIzs7Ow299m5cyfi4+Px6quvIiIiAsOGDcPTTz+Nurq6do9jNpthMpkcHr3J29+cBQDMuyOKAU3Uyzk13FFRUQGr1YqQkBCH7SEhISgtLW1zn4KCAnz77bfQarX4/PPPUVFRgd///ve4cuVKu+PS6enpePHFF50prcf4vugqss9VQimX4bEpQ6Quh4gk5tIXhzeuYSyEaHddY5vNBplMhi1btmDChAmYNWsWVq9ejU2bNrV7Nr18+XIYjUb74+LFi66U6ZXWfN10Fv3LcREciyYi586kg4KCoFAoWp01l5eXtzq7bhEWFoaIiAjodDr7ttjYWAghcOnSJQwd2nrmgkajgUajcaa0HiGvxIQ9P5ZDLgP+4y6eRRORk2fSarUaer0eWVlZDtuzsrKQmJjY5j6TJ09GSUkJqqur7dtOnz4NuVyOyMhIF0ruudbsbTqLnjU6DIMH9JW4GiLyBE4Pd6SlpeH999/Hxo0bkZ+fj2XLlqGoqAipqakAmoYqFixYYG8/f/58BAYGYtGiRcjLy8P+/fvxzDPP4N///d/Rpw//d77F+YoaZBw3AAAen8oF/YmoidPzpFNSUlBZWYlVq1bBYDAgLi4OGRkZiI6OBgAYDAYUFRXZ2/ft2xdZWVl44oknEB8fj8DAQMydOxcvvfRS1/WiB9h4sBBCAFOHD0BsWM+fZkhEHeP0PGkp9PR50sbaRkxK34O6Ris+enQiEm/jcqREPZ1b5kmTe3x0pAh1jVbEhvkjYUig1OUQkQdhSEuswWLDpuxCAMDiO2PancpIRL0TQ1piGccNKDOZMcBPgzljwqQuh4g8DENaQkIIvP9tAQBgYUI0NEqFxBURkadhSEvoSOEVnCg2QauSY/7EaKnLISIPxJCW0Kbs8wCAB8ZHor+vWtpiiMgjMaQlUmaqR2ZeGQBgYcIgaYshIo/FkJbI1iNFsNoE7hgUgOGhflKXQ0QeiiEtAYvVho+PNK3s95tJHIsmovYxpCXwz/xylJrqEeirxr1xoVKXQ0QejCEtgb8fvgAASLkjitPuiOimGNLdrOByNb49WwGZDPj1hLZv3EtE1IIh3c22fNe0QuDU4cGI6u8jcTVE5OkY0t2ovtGKHd9fAgD8ZhLPoono1hjS3eif+WW4VtuIMJ0WU4YFS10OEXkBhnQ3+uRo01n0Q/pIKORc7Y6Ibo0h3U1KrtXhwJnLAJpCmoioIxjS3WRHziUIAUwa3B/Rgb5Sl0NEXoIh3Q1sNoHtOU1DHb/SR0lcDRF5E4Z0N/iu8AqKrtSir0aJmaN5hSERdRxDuhtsP9q0TsecMWHwUTt9g3Yi6sUY0m5mqm9ExgkDAOBX8RzqICLnMKTd7H+PG1DfaMOQAb4YF9VP6nKIyMswpN3si9wSAE13X+GdwInIWQxpNyo11uNwYSUA4OdjwiWuhoi8EUPajXb+UAwhgDsGBXAxJSJyCUPajT5vHur4xdgIiSshIm/FkHaT02VVyDeYoFLIMHt0mNTlEJGXYki7yRe5xQCAKcOCEeCrlrgaIvJWDGk3sNkEvjzWNNRx/zh+YUhErmNIu0FO0VUUX6tDX40S02JDpC6HiLyYSyG9du1axMTEQKvVQq/X48CBAx3a7+DBg1AqlRg7dqwrh/UaLUMd98aFQqvijWaJyHVOh/S2bduwdOlSrFixArm5uUhKSsLMmTNRVFR00/2MRiMWLFiAe+65x+VivYHFasOuE6UAODeaiDrP6ZBevXo1Fi9ejEcffRSxsbF4/fXXERUVhXXr1t10v8ceewzz589HQkKCy8V6g+8Kr6CypgEBPiokDgmUuhwi8nJOhXRDQwNycnKQnJzssD05ORnZ2dnt7vfBBx/g3LlzWLlyZYeOYzabYTKZHB7e4qvjTYsp3RsXCqWCQ/5E1DlOpUhFRQWsVitCQhy/DAsJCUFpaWmb+5w5cwbPPvsstmzZAqWyY8t0pqenQ6fT2R9RUd6xetz1Qx2zODeaiLqAS6d6Ny4UJIRoc/Egq9WK+fPn48UXX8SwYcM6/PrLly+H0Wi0Py5evOhKmd3ucMEVXGke6kgYzKEOIuo8p1agDwoKgkKhaHXWXF5e3ursGgCqqqpw9OhR5Obm4g9/+AMAwGazQQgBpVKJzMxM3H333a3202g00Gg0zpTmETjUQURdzakkUavV0Ov1yMrKctielZWFxMTEVu39/f1x/PhxHDt2zP5ITU3F8OHDcezYMUycOLFz1XsQi9WG3Seb/vGaPZqzOoioazh9L6e0tDQ8/PDDiI+PR0JCAt577z0UFRUhNTUVQNNQRXFxMTZv3gy5XI64uDiH/YODg6HValtt93YtQx39fdWYNLi/1OUQUQ/hdEinpKSgsrISq1atgsFgQFxcHDIyMhAdHQ0AMBgMt5wz3RN9dbzpMvAZozjUQURdRyaEEFIXcSsmkwk6nQ5GoxH+/v5Sl9OKxWrDhJf34EpNA/6+eCLuHBokdUlE5OE6mms85esCR87/NKuDQx1E1JUY0l0g82QZAGBabAiHOoioSzFROkkIgczmWR3Jo0IlroaIehqGdCedKDahxFgPH7UCSRyLJqIuxpDupJa50VOGDeCypETU5RjSndQS0jM41EFEbsCQ7oSCy9U4U14NpVyGqcODpS6HiHoghnQnZOY1zepIGBIInY9K4mqIqCdiSHfCbs7qICI3Y0i7qMxUj9yiawCA5JG82SwRuQdD2kVZzUMd4wb2Q4i/VuJqiKinYki76OsfywE0XWVIROQuDGkX1DVYcfBsBQDgnljO6iAi92FIu+BQQQXMFhvCdVoMD/GTuhwi6sEY0i5oGeq4Oza4zXs7EhF1FYa0k4QQ+Dq/OaRHcKiDiNyLIe2kU2VVKDHWQ6OUI2EwF1QiIvdiSDupZahj8m1B6KPmgkpE5F4MaSe1DHVM5VAHEXUDhrQTrtY04PuiqwA4Hk1E3YMh7YR9py/DJoARoX6I6NdH6nKIqBdgSDvBPvWOZ9FE1E0Y0h1ksdqw9xRDmoi6F0O6g74vugZTvQX9fFQYNzBA6nKIqJdgSHfQnh+bVr27a9gAKOS8ypCIugdDuoO+sV8KzlXviKj7MKQ74OKVWpwuq4ZCLsOUoQOkLoeIehGGdAd80/yFoX5gAO9lSETdiiHdAXvyf1r1joioOzGkb6G2wYJDBZUAOPWOiLofQ/oWss9WosFiQ2RAHwwN7it1OUTUy7gU0mvXrkVMTAy0Wi30ej0OHDjQbtvPPvsM06dPx4ABA+Dv74+EhATs3r3b5YK7257rrjLkAv9E1N2cDult27Zh6dKlWLFiBXJzc5GUlISZM2eiqKiozfb79+/H9OnTkZGRgZycHEydOhVz5sxBbm5up4t3NyHET1PvONRBRBKQCSGEMztMnDgR48ePx7p16+zbYmNjcf/99yM9Pb1DrzFq1CikpKTg+eef71B7k8kEnU4Ho9EIf39/Z8rtlJMlRsx+81v0USmQ+/x0aFVcP5qIukZHc82pM+mGhgbk5OQgOTnZYXtycjKys7M79Bo2mw1VVVXo379/u23MZjNMJpPDQwrfXLfAPwOaiKTgVEhXVFTAarUiJMTxqruQkBCUlpZ26DX++7//GzU1NZg7d267bdLT06HT6eyPqKgoZ8rsMns41EFEEnPpi8Mbv0ATQnToS7WtW7fihRdewLZt2xAc3H7wLV++HEaj0f64ePGiK2V2SmW1GccuXgPAkCYi6SidaRwUFASFQtHqrLm8vLzV2fWNtm3bhsWLF2P79u2YNm3aTdtqNBpoNBpnSutye09dhhDAyDB/hOq0ktZCRL2XU2fSarUaer0eWVlZDtuzsrKQmJjY7n5bt27FI488go8++gizZ892rdJu9nXzpeD38CpDIpKQU2fSAJCWloaHH34Y8fHxSEhIwHvvvYeioiKkpqYCaBqqKC4uxubNmwE0BfSCBQvwxhtvYNKkSfaz8D59+kCn03VhV7pOo9WG/acuA+ANZ4lIWk6HdEpKCiorK7Fq1SoYDAbExcUhIyMD0dHRAACDweAwZ/rdd9+FxWLB448/jscff9y+feHChdi0aVPne+AGR89fRZXZgv6+aoyJ7Cd1OUTUizk9T1oK3T1P+r++ysP6A4V4YHwEVs8d6/bjEVHv45Z50r0FbzhLRJ6CIX2DC5U1OHe5Bkq5DElc4J+IJMaQvkHLWXT8oADo+nCBfyKSFkP6Bi0L/N8zgvcyJCLpMaSvY6pvxHeFTQv8c340EXkChvR19p++jEarwOABvhg8gAv8E5H0GNLX+WdeGQBgeiyHOojIMzCkm1msNnzTfJXhPQxpIvIQDOlmRy9chbGuEQE+Kowf2E/qcoiIADCk7fbkNw11TB0eDKWC/1mIyDMwjdC0HnZW83j0tJEc6iAiz8GQBnDucg3OV9ZCpZAhaWiQ1OUQEdkxpPHTUMekwYHw0/IqQyLyHAxpAP9sDulpnNVBRB6m14f05Sozci5cBcCrDInI8/T6kN51shQ2AYyJ1CEywEfqcoiIHPT6kP7f4wYAwMzRYRJXQkTUWq8O6YpqMw4XNC2oNJshTUQeqFeHdObJMtgEMDpCh6j+HOogIs/Tq0M6wz7UESpxJUREbeu1IX2lpgGHmoc6ZsVxqIOIPFOvDemdx4phtQnERfhjUJCv1OUQEbWp14b09pxLAICHxkdKXAkRUft6ZUjnlZhwssQEtUKOX4yNkLocIqJ29cqQ3p5zEQAwbWQwAnzVEldDRNS+XhfSDRYbvjxWAgD4lT5K4mqIiG6u14X0nvwyXKlpQLCfhsuSEpHH63Uh/cHB8wCAh/SRvAMLEXm8XpVSP1y8hiPnr0Apl2FBwiCpyyEiuqVeFdJrvjkLAPj5mHCE6rQSV0NEdGu9JqRzLlxFZl4Z5DLg91OHSF0OEVGHuBTSa9euRUxMDLRaLfR6PQ4cOHDT9vv27YNer4dWq8XgwYPxzjvvuFSsqxqtNqz6x0kATTM6bgv269bjExG5yumQ3rZtG5YuXYoVK1YgNzcXSUlJmDlzJoqKitpsX1hYiFmzZiEpKQm5ubl47rnnsGTJEuzYsaPTxXfU6/88jR8uGaHro0Ja8rBuOy4RUWfJhBDCmR0mTpyI8ePHY926dfZtsbGxuP/++5Gent6q/Z/+9Cfs3LkT+fn59m2pqan44YcfcOjQoQ4d02QyQafTwWg0wt/fv0P7CCFwqqwKWw4X4f8dvgAAeGPeWF5hSEQeoaO55tSZdENDA3JycpCcnOywPTk5GdnZ2W3uc+jQoVbtZ8yYgaNHj6KxsbHNfcxmM0wmk8PDWUcvXMW9rx+wB/QzM4YzoInI6zgV0hUVFbBarQgJcbyrdkhICEpLS9vcp7S0tM32FosFFRUVbe6Tnp4OnU5nf0RFOX9l4Liofojo1wfTR4Zgw8J4PD71Nqdfg4hIakpXdpLJZA6/CyFabbtV+7a2t1i+fDnS0tLsv5tMJqeDWqmQY+8zd0HFC1aIyIs5FdJBQUFQKBStzprLy8tbnS23CA0NbbO9UqlEYGBgm/toNBpoNBpnSmsTA5qIvJ1TKaZWq6HX65GVleWwPSsrC4mJiW3uk5CQ0Kp9ZmYm4uPjoVKpnCyXiKh3cfpUMy0tDe+//z42btyI/Px8LFu2DEVFRUhNTQXQNFSxYMECe/vU1FRcuHABaWlpyM/Px8aNG7FhwwY8/fTTXdcLIqIeyukx6ZSUFFRWVmLVqlUwGAyIi4tDRkYGoqOjAQAGg8FhznRMTAwyMjKwbNkyrFmzBuHh4XjzzTfx4IMPdl0viIh6KKfnSUvBlXnSRESezC3zpImIqHsxpImIPJhL86S7W8uIjCtXHhIReaKWPLvViLNXhHRVVRUAuHTlIRGRJ6uqqoJOp2v3ea/44tBms6GkpAR+fn43vbLxRi1XKl68eNHrv3BkXzwT++KZvKEvQghUVVUhPDwccnn7I89ecSYtl8sRGRnp8v7+/v4e+0Y5i33xTOyLZ/L0vtzsDLoFvzgkIvJgDGkiIg/Wo0Nao9Fg5cqVXbJYk9TYF8/EvnimntQXr/jikIiot+rRZ9JERN6OIU1E5MEY0kREHowhTUTkwRjSREQezKNDeu3atYiJiYFWq4Ver8eBAwdu2n7fvn3Q6/XQarUYPHgw3nnnnVZtduzYgZEjR0Kj0WDkyJH4/PPPO31cKfqyfv16JCUlISAgAAEBAZg2bRqOHDni0OaFF16ATCZzeISGhnpcXzZt2tSqTplMhvr6+k4dV4q+3HXXXW32Zfbs2fY2nvC+GAwGzJ8/H8OHD4dcLsfSpUvbbOcNn5eO9EXKz0unCQ/18ccfC5VKJdavXy/y8vLEk08+KXx9fcWFCxfabF9QUCB8fHzEk08+KfLy8sT69euFSqUSn376qb1Ndna2UCgU4uWXXxb5+fni5ZdfFkqlUhw+fNjl40rVl/nz54s1a9aI3NxckZ+fLxYtWiR0Op24dOmSvc3KlSvFqFGjhMFgsD/Ky8td7oe7+vLBBx8If39/hzoNBkOnjitVXyorKx36cOLECaFQKMQHH3xgb+MJ70thYaFYsmSJ+PDDD8XYsWPFk08+2aqNt3xeOtIXqT4vXcFjQ3rChAkiNTXVYduIESPEs88+22b7P/7xj2LEiBEO2x577DExadIk++9z584V9957r0ObGTNmiHnz5rl83I5wR19uZLFYhJ+fn/jwww/t21auXCnGjBnjct1tcUdfPvjgA6HT6br0uB3RHe/L3/72N+Hn5yeqq6vt2zzhfbnelClT2gw2b/m8XK+9vtyouz4vXcEjhzsaGhqQk5OD5ORkh+3JycnIzs5uc59Dhw61aj9jxgwcPXoUjY2NN23T8pquHFeqvtyotrYWjY2N6N+/v8P2M2fOIDw8HDExMZg3bx4KCgpc6oe7+1JdXY3o6GhERkbivvvuQ25ubqeOK2VfrrdhwwbMmzcPvr6+Dtulfl86wls+L67ojs9LV/HIkK6oqIDVakVISIjD9pCQEJSWlra5T2lpaZvtLRYLKioqbtqm5TVdOa5UfbnRs88+i4iICEybNs2+beLEidi8eTN2796N9evXo7S0FImJiaisrPSovowYMQKbNm3Czp07sXXrVmi1WkyePBlnzpxx+bhS9eV6R44cwYkTJ/Doo486bPeE96UjvOXz4oru+Lx0FY9eqvTGtaOFEDddT7qt9jdu78hrOnvcjnBHX1q8+uqr2Lp1K/bu3QutVmvfPnPmTPvPo0ePRkJCAoYMGYIPP/wQaWlpLvWjvdo605dJkyZh0qRJ9ucnT56M8ePH46233sKbb77p8nE7wp3vy4YNGxAXF4cJEyY4bPeU96WrXtMT3hdndPfnpbM88kw6KCgICoWi1b+c5eXlrf6FbREaGtpme6VSicDAwJu2aXlNV44rVV9avPbaa3j55ZeRmZmJ22+//aa1+Pr6YvTo0fYzVGe5uy8t5HI57rjjDnud3vi+1NbW4uOPP251Ft0WKd6XjvCWz4szuvPz0lU8MqTVajX0ej2ysrIctmdlZSExMbHNfRISElq1z8zMRHx8PFQq1U3btLymK8eVqi8A8Ne//hV//vOfsWvXLsTHx9+yFrPZjPz8fISFhbnQE/f25XpCCBw7dsxep7e9LwDwySefwGw24ze/+c0ta5HifekIb/m8dFR3f166TPd/V9kxLdNwNmzYIPLy8sTSpUuFr6+vOH/+vBBCiGeffVY8/PDD9vYt06OWLVsm8vLyxIYNG1pNjzp48KBQKBTilVdeEfn5+eKVV15pd0pRe8f1lL785S9/EWq1Wnz66acOU4aqqqrsbZ566imxd+9eUVBQIA4fPizuu+8+4efn53F9eeGFF8SuXbvEuXPnRG5urli0aJFQKpXiu+++6/BxPaUvLe68806RkpLS5nE94X0RQojc3FyRm5sr9Hq9mD9/vsjNzRUnT560P+8tn5eO9EWqz0tX8NiQFkKINWvWiOjoaKFWq8X48ePFvn377M8tXLhQTJkyxaH93r17xbhx44RarRaDBg0S69ata/Wa27dvF8OHDxcqlUqMGDFC7Nixw6njekpfoqOjBYBWj5UrV9rbpKSkiLCwMKFSqUR4eLh44IEHHP7iekpfli5dKgYOHCjUarUYMGCASE5OFtnZ2U4d11P6IoQQp06dEgBEZmZmm8f0lPelrb8/0dHRDm285fNyq75I+XnpLK4nTUTkwTxyTJqIiJowpImIPBhDmojIgzGkiYg8GEOaiMiDMaSJiDwYQ5qIyIMxpImIPBhDmojIgzGkiYg8GEOaiMiD/X/pwFSaFIa9YQAAAABJRU5ErkJggg==", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "print(\"Respuesta a un escalón\")\n", "y,t = step(SSI)\n", "plt.plot(t,y);" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Implementación\n", "Para implementar en Arduino el controlador en espacio de estados debemos considerar:" ] }, { "cell_type": "markdown", "metadata": { "id": "O-UIDAoxXBtO", "slideshow": { "slide_type": "-" } }, "source": [ "![Lazo Cerrado con Integrador](lazo-cerrado-con-integrador.png)" ] }, { "cell_type": "markdown", "metadata": { "id": "pbGjnzCSWvid", "slideshow": { "slide_type": "-" } }, "source": [ "Del diagrama de bloque tenemos que la acción $u$ es:\n", "\n", "$$u = K_e\\int_0^t e(\\tau)\\,d\\tau - k \\,x(t)$$ \n", "\n", "con $e(t)=r-y(t)$ y $x(t) = C^{-1}\\, y(t)$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "Esta ecuación se puede programar de la siguiente forma: " ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "vr_zCE0k-HLJ", "outputId": "d56f527a-c2f8-4401-9646-96f7623ba238", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "float ke = 80.848605;\n", "float kp = 0.652756;\n", "\n", "float y = 0;\n", "float u = 0;\n", "float ref = 500;\n", "float error = 0;\n", "float eSum = 0;\n", "float dt = 0.001;\n", "\n", "void control_loop() {\n", "\n", " error = ref - y;\n", " eSum += error * dt;\n", " u = ke * eSum + kp * y;\n", "\n", "}\n", "\n" ] } ], "source": [ "nke = ks[ke]\n", "nkp = ks[k]/SSI.C[0,0]\n", "\n", "print(\"\"\"\n", "float ke = %f;\n", "float kp = %f;\n", "\n", "float y = 0;\n", "float u = 0;\n", "float ref = 500;\n", "float error = 0;\n", "float eSum = 0;\n", "float dt = 0.001;\n", "\n", "void control_loop() {\n", "\n", " error = ref - y;\n", " eSum += error * dt;\n", " u = ke * eSum + kp * y;\n", "\n", "}\n", "\"\"\" % (nke,nkp))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "[código arduino lazo cerrado](lazo_cerrado/lazo_cerrado.ino) para el espacio de estados." ] }, { "cell_type": "markdown", "metadata": { "id": "S-W_AtY1k841", "slideshow": { "slide_type": "subslide" } }, "source": [ "La implemntación del controlador en espacio de estado para este sistema es similar al controlador PID, luego podemos usar los mismos valores: " ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 452 }, "id": "BhPqG1rgdoEC", "outputId": "c351bf42-33c0-4837-bf40-c0061d005a7b", "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Controlador PI :\n" ] }, { "data": { "text/latex": [ "$$\\frac{0.65 s + 81}{s}$$" ], "text/plain": [ "TransferFunction(array([ 0.65, 81. ]), array([1, 0]))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Kp = 0.65 \n", "Ki = 81 \n", "K = Kp + Ki/s \n", "\n", "print(\"Controlador PI :\")\n", "display(K)\n", "CPID = feedback(K*GR,1)\n", "yy,tt = step(CPID)\n", "plt.plot(tt,yy); " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Slideshow", "colab": { "provenance": [] }, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.6" }, "title": "Modelizado Experimental", "week": 10 }, "nbformat": 4, "nbformat_minor": 1 }