{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "## Libreria de control\n",
    "try:\n",
    "  from control.matlab import *\n",
    "except:\n",
    "  !pip install control\n",
    "  from control.matlab import *\n",
    "    \n",
    "## Libreria para graphicar\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy\n",
    "\n",
    "## Libreria para calculo simbolico\n",
    "import sympy\n",
    "\n",
    "## Libreria de control\n",
    "import control\n",
    "\n",
    "## Libreria para widgets\n",
    "import ipywidgets as widgets\n",
    "\n",
    "## Libreria para animaciones\n",
    "from matplotlib.animation import FuncAnimation\n",
    "plt.rcParams[\"animation.html\"] = \"html5\"#\"jshtml\"\n",
    "\n",
    "## Libreria para importar Iframe de Youtube\n",
    "from IPython.display import IFrame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad de sistemas LTI\n",
    "_Entendamos como un sistema se comporta_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Dinámica del sistema\n",
    "\n",
    "![](3-ecuacion.png)\n",
    "\n",
    "- Empecemos ignorando la entrada del sistema, el estado será $x$\n",
    "\n",
    "$$\\array{\\dot{x}=A\\, x &&& x(t_0)=x_0}$$\n",
    "\n",
    "¿Cuál es la respuesta temporal?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Resolviendo la ecuación diferencial ordinaria\n",
    "\n",
    "Si todo fuera escalar:\n",
    "\n",
    "$$\\array{\\dot{x}=a\\, x  &,& x(t_0)=x_0 & \\to & x(t)=e^{a(t-t_0))}x_0 }$$\n",
    "\n",
    "Como sabemos esto? verificamos la posible solución:\n",
    "\n",
    "$$\\array{\n",
    "x(t_0)=e^{a(t_0-t_0))}x_0=e^{0}x_0=x_0 & \\text{condición inicial verificada}\\\\\n",
    "\\frac{d}{dt}x(t_0)=ae^{a(t-t_0))}x_0=ax & \\text{dinámica verificada}\n",
    "} $$\n",
    "\n",
    "Para sistemas de orden superior, tenemos una versión matricial de esto:\n",
    "\n",
    "$$\\array{\\dot{x}=A\\, x &,& x(t_0)=x_0 &\\to& x(t)=e^{A(t-t_0))}x_0}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Exponencial de una matriz\n",
    "\n",
    "La definición es similar a la definición de escalares: \n",
    "\n",
    "$$e^{at}=\\sum_{k=0}^\\infty\\frac{a^kt^k}{k!} \\qquad\\qquad e^{At}=\\sum_{k=0}^\\infty\\frac{A^kt^k}{k!}$$\n",
    "\n",
    "cuya derivada es:\n",
    "\n",
    "$$\\frac{d}{dt} \\sum_{k=0}^\\infty\\frac{A^kt^k}{k!} \n",
    "= 0 + \\sum_{k=0}^\\infty\\frac{kA^kt^{k-1}}{k!} \n",
    "= A\\sum_{k=0}^\\infty\\frac{kA^{k-1}t^{k-1}}{(k-1)!}\n",
    "= A\\sum_{k=0}^\\infty\\frac{A^kt^k}{k!} $$\n",
    "\n",
    "es decir que:\n",
    "\n",
    "$$\\frac{d}{dt}e^{At}=Ae^{At}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Resolviendo la ecuación del controlador\n",
    "\n",
    "La exponencial de una matriz tiene un papel fundamental y por eso tiene su propio nombre. \n",
    "\n",
    "**Matriz de transición de estado**\n",
    "\n",
    "$$e^{A(t-t_0))}=\\Phi(t,t_0)$$\n",
    "\n",
    "para una dinámica de estado $\\dot{x}= A x$ la solución será $x(t)=\\Phi(t,\\tau)x(\\tau)$, tiene las siguientes propiedades: \n",
    "\n",
    "$$\\cases{\\frac{d}{dt}\\Phi(t,t_0)=A\\Phi(t,t_0) \\\\ \\Phi(t,t)=I}$$\n",
    "\n",
    "Cuando tenemos un sistema controlado, tendremos:\n",
    "\n",
    "$$x(t)= \\Phi(t,t_0)x(t_0)+\\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Resolviendo la ecuación del controlador\n",
    "\n",
    "Verifiquemos que la ecuación cumple con la condición inicial:\n",
    "\n",
    "$$x(t_0)= \\Phi(t_0,t_0)x(t_0)+\\int_{t_0}^{t_0}\\Phi(t_0,\\tau)B\\,u(\\tau)d\\tau = I x(t_0) + 0 = x(t_0)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "$$x(t)= \\Phi(t,t_0)x(t_0)+\\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "y cumple para la derivada: \n",
    "\n",
    "$$\\frac{d}{dt}x(t)= A\\Phi(t,t_0)x(t_0)+\\frac{d}{dt}\\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "$$\\frac{d}{dt}\\int_{t_0}^{t}f(t,\\tau)d\\tau = f(t,t) +\\int_{t_0}^{t} \\frac{d}{dt}f(t,\\tau)d\\tau = \\Phi(t,t)B\\,u(t) + \\int_{t_0}^{t} A\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "$$\\frac{d}{dt}x(t)= A\\Phi(t,t_0)x(t_0)+B\\,u(t) + \\int_{t_0}^{t} A\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "$$\\frac{d}{dt}x = Ax + B u$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# En resumen\n",
    "\n",
    "tenemos que\n",
    "\n",
    "$$\\dot{x}=A\\,x+B\\,u \\qquad y \\qquad y=C\\,x$$\n",
    "\n",
    "luego,\n",
    "\n",
    "$$y(t) = C \\Phi(t,t_0)x(t_0) + C \\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "sabiendo que \n",
    "\n",
    "$$\\Phi(t,\\tau)= e^{A(t-\\tau)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad\n",
    "\n",
    "Lo primero que debemos hacer es entender por que un sistema explota o no.\n",
    "\n",
    "Recordemos que los objetivos en control son: \n",
    "     \n",
    "- Ser estable\n",
    "- Seguir el objetivo\n",
    "- Ser robusto\n",
    "- Entre otros"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en sistemas escalares\n",
    "\n",
    "- Es útil empezar con sistemas escalares para ganar intuición.\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "con $a>0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.linspace(0,5,num=100)\n",
    "y = numpy.exp(1*t)\n",
    "plt.plot(t,y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en sistemas escalares\n",
    "\n",
    "- Es útil empezar con sistemas escalares para ganar intuición.\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "con $a<0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.linspace(0,5,num=100)\n",
    "y = numpy.exp(-1*t)\n",
    "plt.plot(t,y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en sistemas escalares\n",
    "\n",
    "- Es útil empezar con sistemas escalares para ganar intuición.\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "con $a=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.linspace(0,5,num=100)\n",
    "y = numpy.exp(0*t)\n",
    "plt.plot(t,y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Tres casos\n",
    "\n",
    "- Asintóticamente estable: $x(t)\\to0, \\forall x(0)$\n",
    "- Inestable: $\\exists x(0) : ||{x(t)}||\\to\\infty$\n",
    "- Críticamente estable: en medio, no explota pero tampoco llega a cero\n",
    "\n",
    "**De**\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "tenemos entonces:\n",
    "\n",
    "$$\\cases{\n",
    "a>0 \\quad: & \\text{inestable} \\\\\n",
    "a<0 \\quad: & \\text{asintóticamente estable}\\\\\n",
    "a=0 \\quad: & \\text{críticamente estable}\n",
    "}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en Matrices\n",
    "\n",
    "$$\\dot{x}=A\\,x \\quad \\to\\quad x(t)= e^{At}x(0)$$\n",
    "\n",
    "No podemos decir que $A>0$, pero podemos usar los valores propios.\n",
    "\n",
    "$$A\\,v = \\lambda\\,v$$\n",
    "\n",
    "donde $\\lambda \\in \\mathscr{C}$ son los valores propios y $v \\in \\mathscr{R}^n$ son los vectores propios. \n",
    "\n",
    "Los valores propios no diran como la matriz $A$ actua en cada dirección. \n",
    "\n",
    "**En MATLAB**\n",
    "\n",
    "``` matlab\n",
    ">> eig(A)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Ejemplo\n",
    "\n",
    "Para la siguiente matriz \n",
    "\n",
    "$$A = \\left[\\array{1&0\\\\0&-1}\\right]$$\n",
    "\n",
    "encontrar los valores y vectores propios ⚠️"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\array{\\lambda_1=1 &,& \\lambda_2=-1 &,& v_1=\\left[\\array{1\\\\0}\\right] &,& v_1=\\left[\\array{0\\\\1}\\right]}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad de Matrices\n",
    "\n",
    "$$\\dot{x}=A\\,x \\quad \\to\\quad x(t)= e^{At}x(0)$$\n",
    "\n",
    "- Asintóticamente estable (si y solo si):\n",
    "  $$Re(\\lambda)<0, \\forall\\lambda \\in \\texttt{eig}(A)$$\n",
    "- Inestable (si):\n",
    "  $$\\exists\\lambda\\in\\texttt{eig}(A) : Re(\\lambda)>0$$\n",
    "- Críticamente estable (solo si):\n",
    "  $$Re(\\lambda)\\le0, \\forall\\lambda \\in \\texttt{eig}(A)$$\n",
    "- Críticamente estable (si):\n",
    "  Un valor propio es cero y el resto tienen parte real negativa **o** Dos valores propios son puramente imaginarios y el resto tienen parte real negativa."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# El cuento de dos pendulos\n",
    "\n",
    "Encontremos las matrices para un péndulo regular y un péndulo invertido. Sin fricción. ⚠️"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$A_{\\text{péndulo regular}} = \\left[\\array{0&1\\\\-1&0}\\right] \\qquad A_{\\text{péndulo invertido}} = \\left[\\array{0&1\\\\1&0}\\right]$$\n",
    "\n",
    "encontremos los valores propios ⚠️"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Para el péndulo regular tenemos:\n",
    "  $$\\lambda_1 = j \\qquad \\lambda_2=-j$$\n",
    "  \n",
    "- Para el péndulo invertido tenemos:\n",
    "  $$\\lambda_1 = -1 \\qquad \\lambda_2=1$$\n",
    "  \n",
    "¿Sistemas estables? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Enjambre de robots\n",
    "\n",
    "Analicemos la estabilidad de un enjambre de robots para resolver el problema de _Rendezvous_\n",
    "\n",
    "- Tenemos una colección de robots que pueden medir su posición relativa a sus vecinos. \n",
    "- Problema: hacer que todos los robots se encuentren en el mismo lugar (no especificada). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Caso de dos robots simples\n",
    "\n",
    "Anteriormente revisamos el caso de dos robots en una linea. \n",
    "\n",
    "![](3-dos-robots.png)\n",
    "\n",
    "Si los robots se dirigen el uno hacia el otro tenemos:\n",
    "\n",
    "$$\\cases{u_1=x_2-x_1\\\\u_2=x_1-x_2}$$\n",
    "\n",
    "La matriz dinámica sera entonces:\n",
    "\n",
    "$$\\dot{x}=\\left[\\array{-1&1\\\\1&-1}\\right]x$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en el caso de dos robots simples\n",
    "\n",
    "$$A = \\left[\\array{-1&1\\\\1&-1}\\right] \\qquad \\lambda_1 = 0 \\qquad \\lambda_2=-2$$\n",
    "\n",
    "En este sistema tenemos un valor propio 0 y todos los demas tienen parte real negativa. Aqui el estado del sistema terminara en algo llamado el ***Espacio nulo (null-space)*** de $A$:\n",
    "\n",
    "$$null(A)=\\{x: Ax = 0\\}$$\n",
    "\n",
    "para el caso particular de esta $A$, el espacio nulo es:\n",
    "\n",
    "$$null(A)=\\{x: x = \\left[\\array{\\alpha\\\\\\alpha}\\right] , \\alpha \\in \\mathscr{R} \\}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en el caso de dos robots simples\n",
    "\n",
    "Si $x_1\\to\\alpha$ y $x_2\\to\\alpha$ entonces tenemos que $(x_1-x_2)\\to0$ por lo que el _Rendezvous_ se logró."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dosRobots = ss([[-1,1],[1,-1]],[[0],[0]],[[1,0],[0,1]],[[0],[0]])\n",
    "yout, T = step(dosRobots,X0=[-10,10])\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.plot([-12,12],[0,0],color='gray')\n",
    "ax.axis('off')\n",
    "#ax.plot([0,0],[-1,1],color='r')\n",
    "plt.xlim(-12,12);\n",
    "l, = ax.plot([-10],[0],color='b',marker='.',markersize=30)\n",
    "r, = ax.plot([10],[0],color='r',marker='.',markersize=30)\n",
    "\n",
    "def animate(i):\n",
    "    l.set_data([yout[i,0]], [0])\n",
    "    r.set_data([yout[i,1]], [0])\n",
    "\n",
    "ani = FuncAnimation(fig, animate, frames=len(T), interval=1000/40);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x1e92b0d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(ani)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Si hay más de dos robots, deberiamos pensar en llevarlos a todos al centroide de sus vecinos (o algo parecido)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "# El caso con multiples robots\n",
    "\n",
    "$$\\dot{x}_i = \\sum_{j\\in N_i}(x_j-x_i) \\qquad \\dot{x}=-Lx$$\n",
    "\n",
    "El grafo generado est..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Realimentación del sistema en espacio de estados\n",
    "\n",
    "Sabemos que lo primero que debemos hacer para controlar un sistema es llevarlo a ser asintóticamente estable. Es decir que la parte real de todos sus valores propios sea negativa. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Una particula en una linea recta sin fricción\n",
    "\n",
    "Diseñemos un control proporcional para el sistema propuesto, utilizando el espacio de estados.\n",
    "\n",
    "$$m \\ddot{x} = F$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "La ecuación de estado quedaría asi (con $m=1$). \n",
    "\n",
    "$$\\dot{\\mathbf{x}}=\\left[\\begin{array}{}0&1\\\\0&0\\end{array}\\right]\\mathbf{x}+\\left[\\begin{array}{}0\\\\1\\end{array}\\right]\\mathbf{u}$$\n",
    "\n",
    "La ecuación de salida sería.\n",
    "\n",
    "$$\\mathbf{y}=\\left[\\begin{array}{}1&0\\end{array}\\right]\\mathbf{x}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Control de la particula\n",
    "\n",
    "Si queremos controlar el sistema con un lazo cerrado debemos conectar de alguna forma la salida $\\mathbf{y}$ con la entrada $\\mathbf{u}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAAD4CAYAAADy46FuAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAXJElEQVR4nO3de5Cd9X3f8fenIOThklhYAitcDDSqbTzuyO56g3AnpRhsYDqIJE4MnY7lFEZ2a9q4doLloWNnSDLFZAaYTEhjxRDjxuZiUtdKI4pBQP2HBWihMuISLFm4RSMNUsDXiugC3/5xHjWHZc+zuzpn9+ya92vmzHme3+WcL4+W/exzOedJVSFJUi//YNgFSJLmNoNCktTKoJAktTIoJEmtDApJUqsjh13A4Vi8eHGddtppwy5DkuaVRx999G+rasl0583LoDjttNMYGxsbdhmSNK8k+d+HM89DT5KkVgaFJKmVQSFJamVQSJJaGRSSpFYDCYoktyTZneSJHv1J8kdJtiV5PMm7u/pWJdnaPFYNoh5J0uAMao/iS8AFLf0XAsuax2rgPwMkOR74HPBLwCjwuSSLBlSTJGkABhIUVfUt4MWWISuBL1fHQ8AbkywFPgDcW1UvVtUPgHtpDxxp7vrEJzoP6WfMbH3g7iTgua71HU1br/bXSLKazt4Ip5566sxUKfVj8+ZhVyDNiNk6mZ0J2qql/bWNVWuraqSqRpYsmfYn0CVJh2m2gmIHcErX+snAzpZ2SdIcMVtBsQ74cHP101nAj6pqF3AP8P4ki5qT2O9v2iRJc8RAzlEkuQ04B1icZAedK5kWAFTVnwLrgYuAbcBe4DebvheT/B6wqXmpa6qq7aS4JGmWDSQoquqySfoL+HiPvluAWwZRhyRp8PxktiSplUEhSWplUEiSWhkUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgaFJKmVQSFJamVQSJJaGRSSpFYGhSSplUEhSWplUEiSWhkUkqRWAwmKJBckeSbJtiRrJui/Icnm5vHdJD/s6nu5q2/dIOqRJA1O33e4S3IEcBNwPrAD2JRkXVU9dWhMVf2HrvH/DnhX10u8VFXL+61DkjQzBrFHMQpsq6rtVbUfuB1Y2TL+MuC2AbyvJGkWDCIoTgKe61rf0bS9RpK3AKcD93c1vyHJWJKHklzS602SrG7Gje3Zs2cAZUuSpmIQQZEJ2qrH2EuBu6rq5a62U6tqBPiXwI1J/uFEE6tqbVWNVNXIkiVL+qtYkjRlgwiKHcApXesnAzt7jL2UcYedqmpn87wdeJBXn7+QJA3ZIIJiE7AsyelJjqITBq+5einJW4FFwMautkVJFjbLi4H3Ak+NnytJGp6+r3qqqoNJrgTuAY4AbqmqJ5NcA4xV1aHQuAy4vaq6D0u9HfhCklfohNa13VdLSZKGr++gAKiq9cD6cW2fHbf+uxPM+zbwzkHUIEmaGX4yW5LUyqCQJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrQwKSVKrgQRFkguSPJNkW5I1E/R/JMmeJJubxxVdfauSbG0eqwZRjyRpcPq+w12SI4CbgPOBHcCmJOsmuKXpHVV15bi5xwOfA0aAAh5t5v6g37okSYMxiD2KUWBbVW2vqv3A7cDKKc79AHBvVb3YhMO9wAUDqEmSNCCDCIqTgOe61nc0beP9WpLHk9yV5JRpziXJ6iRjScb27NkzgLIlSVMxiKDIBG01bv2vgNOq6h8D9wG3TmNup7FqbVWNVNXIkiVLDrtYSdL0DCIodgCndK2fDOzsHlBVL1TVvmb1z4B/MtW5kqThGkRQbAKWJTk9yVHApcC67gFJlnatXgw83SzfA7w/yaIki4D3N22SpDmi76uequpgkivp/II/Arilqp5Mcg0wVlXrgH+f5GLgIPAi8JFm7otJfo9O2ABcU1Uv9luTJGlwUjXhKYE5bWRkpMbGxoZdhvRq55zTeX7wwWFWIfWU5NGqGpnuPD+ZLUlqZVBIkloZFJKkVgaFJKmVQSFJamVQSJJaGRSSpFYGhSSplUEhSWplUEiSWhkUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgMJiiQXJHkmybYkaybo/2SSp5I8nmRDkrd09b2cZHPzWDd+riRpuPq+w12SI4CbgPPp3AN7U5J1VfVU17D/BYxU1d4k/wa4DvhQ0/dSVS3vtw5J0swYxB7FKLCtqrZX1X7gdmBl94CqeqCq9jarDwEnD+B9JUmzYBBBcRLwXNf6jqatl8uBu7vW35BkLMlDSS7pNSnJ6mbc2J49e/qrWJI0ZX0fegIyQduEN+JO8q+AEeCfdTWfWlU7k5wB3J9kS1V97zUvWLUWWAude2b3X7YkaSoGsUexAzila/1kYOf4QUnOA64GLq6qfYfaq2pn87wdeBB41wBqkiQNyCCCYhOwLMnpSY4CLgVedfVSkncBX6ATEru72hclWdgsLwbeC3SfBJckDVnfh56q6mCSK4F7gCOAW6rqySTXAGNVtQ74Q+BY4GtJAP5PVV0MvB34QpJX6ITWteOulpIkDdkgzlFQVeuB9ePaPtu1fF6Ped8G3jmIGiRJM8NPZkuSWhkUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgaFJKmVQSFJamVQSJJaGRSSpFYGhSSp1UC+60nzT1WxceNGHnnkEX7yk59w3HHHMTo6yooVK2i+uFGSAIPidefAgQPcfPPNXHfddezevZsDBw5w4MABFixYwIIFCzjhhBO46qqruPzyy1mwYMGwy5U0BxgUryM//elPufDCC3nsscfYu3fvq/r279/P/v37efbZZ/nUpz7FV7/6VdavX8+xxx47pGolzRWeo3idOHDgABdeeCGbNm16TUiMt3fvXh555BEuuugiDhw4MEsVSpqrDIrXiZtvvpnHHnuMffv2TT4Y2LdvH48++ii33HLLDFcmaa4bSFAkuSDJM0m2JVkzQf/CJHc0/Q8nOa2r7zNN+zNJPjCIevRqVcV111036Z7EeHv37uW6666jqmaoMknzQd9BkeQI4CbgQuBM4LIkZ44bdjnwg6r6ReAG4PPN3DPp3GP7HcAFwJ80r6cB2rhxI7t375584ASef/55Nm7cOOCKJM0ngziZPQpsq6rtAEluB1YC3fe+Xgn8brN8F/DH6VyDuRK4var2Ac8m2da8Xutvpu17/i8f+oK/vKbquxvu5KV9+w9r7kv79vPxG+/kH23xktnJfHbnjwG4xp9N/YwZxKGnk4DnutZ3NG0Tjqmqg8CPgDdNcS4ASVYnGUsy5gnW6Tnwd3t55eDBw5r7ysGDHNw3vUNWkn62DGKPYqI/Nccf1O41ZipzO41Va4G1ACMjI3XHR1dMp8bXtRtfephP37OA/funv1excOFRXHHuO/gtt/fkbvs5APzZ1Fx158cOb94g9ih2AKd0rZ8M7Ow1JsmRwM8DL05xrvo0Ojp62B+eO/LII3nPe94z4IokzSeDCIpNwLIkpyc5is7J6XXjxqwDVjXLHwTur86lNOuAS5urok4HlgGPDKAmdVmxYgUnnHDCYc098cQTWbHCv5Cl17O+g6I553AlcA/wNHBnVT2Z5JokFzfDbgbe1Jys/iSwppn7JHAnnRPf/wP4eFW93G9NerUkXHXVVRx99NHTmnf00Udz1VVX+d1P0utc5uM18iMjIzU2NjbsMuaVAwcOcO6557Jp06Ypfehu4cKFjI6OsmHDBr/zaarOOafz/OCDw6xC6inJo1U1Mt15fjL7dWLBggXcfffdjI6OTrpncfTRRzM6Osr69esNCUkGxevJsccey4YNG7j++us544wzOOaYY1i4cCFJWLhwIccccwxnnHEG119/PRs2bPALASUBfnvs686CBQv46Ec/yurVq9m4cSObNm161f0ozjrrLM9JSHoVg+J1Kglnn302Z5999rBLkTTHeehJktTKoJAktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSq76CIsnxSe5NsrV5XjTBmOVJNiZ5MsnjST7U1felJM8m2dw8lvdTjyRp8Prdo1gDbKiqZcCGZn28vcCHq+odwAXAjUne2NX/O1W1vHls7rMeSdKA9RsUK4Fbm+VbgUvGD6iq71bV1mZ5J7AbWNLn+0qSZkm/QXFiVe0CaJ5PaBucZBQ4CvheV/MfNIekbkiysGXu6iRjScb27NnTZ9mSpKmaNCiS3JfkiQkeK6fzRkmWAv8F+M2qeqVp/gzwNuA9wPHAp3vNr6q1VTVSVSNLlrhDIkmzZdI73FXVeb36kjyfZGlV7WqCYHePcT8H/DXwH6vqoa7X3tUs7kvy58BvT6t6SdKM6/fQ0zpgVbO8CvjG+AFJjgK+Dny5qr42rm9p8xw65zee6LMeSdKA9RsU1wLnJ9kKnN+sk2QkyRebMb8B/DLwkQkug/1Kki3AFmAx8Pt91iNJGrBJDz21qaoXgPdN0D4GXNEs/wXwFz3mn9vP+0uSZp6fzJYktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAktTIoJEmtDApJUqu+giLJ8UnuTbK1eV7UY9zLXTctWtfVfnqSh5v5dzR3w5MkzSH97lGsATZU1TJgQ7M+kZeqannzuLir/fPADc38HwCX91mPJGnA+g2KlcCtzfKtdO57PSXNfbLPBe46nPmSpNnRb1CcWFW7AJrnE3qMe0OSsSQPJTkUBm8CflhVB5v1HcBJvd4oyermNcb27NnTZ9mSpKma9J7ZSe4D3jxB19XTeJ9Tq2pnkjOA+5NsAX48wbjq9QJVtRZYCzAyMtJznCRpsCYNiqo6r1dfkueTLK2qXUmWArt7vMbO5nl7kgeBdwF/CbwxyZHNXsXJwM7D+G+QJM2gfg89rQNWNcurgG+MH5BkUZKFzfJi4L3AU1VVwAPAB9vmS5KGq9+guBY4P8lW4PxmnSQjSb7YjHk7MJbkO3SC4dqqeqrp+zTwySTb6JyzuLnPeiRJAzbpoac2VfUC8L4J2seAK5rlbwPv7DF/OzDaTw2SpJnlJ7MlSa0MCklSK4NCktTKoJAktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAkteorKJIcn+TeJFub50UTjPnnSTZ3Pf4uySVN35eSPNvVt7yfeiRJg9fvHsUaYENVLQM2NOuvUlUPVNXyqloOnAvsBb7ZNeR3DvVX1eY+65EkDVi/QbESuLVZvhW4ZJLxHwTurqq9fb6vJGmW9BsUJ1bVLoDm+YRJxl8K3Dau7Q+SPJ7khiQLe01MsjrJWJKxPXv29Fe1JGnKJg2KJPcleWKCx8rpvFGSpcA7gXu6mj8DvA14D3A88Ole86tqbVWNVNXIkiVLpvPWkqQ+HDnZgKo6r1dfkueTLK2qXU0Q7G55qd8Avl5VB7pee1ezuC/JnwO/PcW6JUmzpN9DT+uAVc3yKuAbLWMvY9xhpyZcSBI65zee6LMeSdKA9RsU1wLnJ9kKnN+sk2QkyRcPDUpyGnAK8D/Hzf9Kki3AFmAx8Pt91iNJGrBJDz21qaoXgPdN0D4GXNG1/n3gpAnGndvP+0uSZp6fzJYktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAktTIoJEmtDApJUiuDQpLUqq+gSPLrSZ5M8kqSkZZxFyR5Jsm2JGu62k9P8nCSrUnuSHJUP/VIkgav3z2KJ4BfBb7Va0CSI4CbgAuBM4HLkpzZdH8euKGqlgE/AC7vsx5J0oD1FRRV9XRVPTPJsFFgW1Vtr6r9wO3AyiQBzgXuasbdClzSTz2SpMHr657ZU3QS8FzX+g7gl4A3AT+sqoNd7a+5r/YhSVYDqwFOPfXUmalU6sfy5cOuQJoRkwZFkvuAN0/QdXVVfWMK75EJ2qqlfUJVtRZYCzAyMtJznDQ0N9447AqkGTFpUFTVeX2+xw7glK71k4GdwN8Cb0xyZLNXcahdkjSHzMblsZuAZc0VTkcBlwLrqqqAB4APNuNWAVPZQ5EkzaJ+L4/9lSQ7gBXAXye5p2n/hSTrAZq9hSuBe4CngTur6snmJT4NfDLJNjrnLG7upx5J0uCl84f9/DIyMlJjY2PDLkOS5pUkj1ZVz8+89eInsyVJrQwKSVIrg0KS1MqgkCS1mpcns5P8BJjsq0PmgsV0Pi8y182HOudDjWCdg2adg/XWqjpuupNm4ys8ZsIzh3PmfrYlGbPOwZgPNYJ1Dpp1DlaSw7pc1ENPkqRWBoUkqdV8DYq1wy5giqxzcOZDjWCdg2adg3VYdc7Lk9mSpNkzX/coJEmzxKCQJLWaF0GR5A+T/E2Sx5N8Pckbe4y7IMkzSbYlWTOEOn89yZNJXknS81K5JN9PsiXJ5sO9XK0f06hzaNszyfFJ7k2ytXle1GPcy8123Jxk3SzW17ptkixMckfT/3CS02artnF1TFbnR5Ls6dqGVwyhxluS7E7yRI/+JPmj5r/h8STvnu0amzomq/OcJD/q2pafHUKNpyR5IMnTzf/jvzXBmOlvz6qa8w/g/cCRzfLngc9PMOYI4HvAGcBRwHeAM2e5zrcDbwUeBEZaxn0fWDzE7TlpncPensB1wJpmec1E/+ZN30+HsP0m3TbAvwX+tFm+FLhjjtb5EeCPZ7u2cTX8MvBu4Ike/RcBd9O5K+ZZwMNztM5zgP8+5G25FHh3s3wc8N0J/s2nvT3nxR5FVX2z/v7e2g/RuRveeKPAtqraXlX7gduBlbNVI0BVPV1Vc/4T41Osc9jbcyVwa7N8K3DJLL73ZKaybbrrvwt4X5KJbv87k4b9bzglVfUt4MWWISuBL1fHQ3TujLl0dqr7e1Ooc+iqaldVPdYs/4TOPYBOGjds2ttzXgTFOP+aThqOdxLwXNf6Dl67geaKAr6Z5NEkq4ddTA/D3p4nVtUu6PzwAyf0GPeGJGNJHkoyW2EylW3z/8c0f+T8iM7NuWbTVP8Nf605BHFXklMm6B+2Yf8sTseKJN9JcneSdwyzkOZw57uAh8d1TXt7zpmv8EhyH/DmCbqurqpvNGOuBg4CX5noJSZoG/i1v1OpcwreW1U7k5wA3Jvkb5q/VgZmAHXO+PZsq3EaL3Nqsy3PAO5PsqWqvjeYCnuayraZlZ/HSUylhr8CbquqfUk+Rmcv6NwZr2x65sK2nIrHgLdU1U+TXAT8N2DZMApJcizwl8AnqurH47snmNK6PedMUFTVeW39SVYB/wJ4XzUH2sbZAXT/NXQysHNwFXZMVucUX2Nn87w7ydfpHCIYaFAMoM4Z355tNSZ5PsnSqtrV7Bbv7vEah7bl9iQP0vkLaqaDYirb5tCYHUmOBH6e2T9sMWmdVfVC1+qf0TkHONfMyv/b/er+hVxV65P8SZLFVTWrXxaYZAGdkPhKVf3XCYZMe3vOi0NPSS6gc3/ti6tqb49hm4BlSU5PchSdE4izdhXMVCU5Jslxh5bpnKif8CqKIRv29lwHrGqWVwGv2QtKsijJwmZ5MfBe4KlZqG0q26a7/g8C9/f4A2cmTVrnuGPTF9M5pj3XrAM+3Fytcxbwo0OHJeeSJG8+dB4qySid368vtM8aeA0Bbgaerqrrewyb/vYc5hn6aZzJ30bnmNrm5nHoapJfANaPO5v/XTp/UV49hDp/hU5a7wOeB+4ZXyedK1C+0zyenKt1Dnt70jmevwHY2jwf37SPAF9sls8GtjTbcgtw+SzW95ptA1xD548ZgDcAX2t+dh8Bzpjtf+cp1vmfmp/D7wAPAG8bQo23AbuAA83P5eXAx4CPNf0Bbmr+G7bQckXhkOu8smtbPgScPYQa/ymdw0iPd/2+vKjf7elXeEiSWs2LQ0+SpOExKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSq/8HX+pcBXe2YDsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot([-2,2],[0,0])\n",
    "plt.plot([0,0],[-1,1],color='r')\n",
    "plt.plot([-1],[0],color='k',marker='.',markersize=30)\n",
    "plt.xlim(-2,2);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "El objetivo de control es que se mueva al origen. **¿Cómo lo logramos?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\left.\\begin{array}{}u>0 \\text{ si } y<0 \\\\ u<0 \\text{ si } y>0\\end{array}\\right\\}\\qquad \\to \\qquad u=-y$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Modificación de la dinamica del sistema \n",
    "\n",
    "En general tenemos: \n",
    "\n",
    "$$\\mathbf{u}=-K\\mathbf{y} = -KC\\mathbf{x} $$\n",
    "\n",
    "entonces: \n",
    "\n",
    "$$\\dot{\\mathbf{x}}=A\\mathbf{x}+B\\mathbf{u}=A\\mathbf{x}-BKC\\mathbf{x} = \\left(A-BKC\\right)\\mathbf{x}$$\n",
    "\n",
    "tenemos aquí un nuevo sistema, el sistema en lazo cerrado.\n",
    "\n",
    "$$\\dot{\\mathbf{x}} = \\left(A-BKC\\right)\\mathbf{x}=\\hat{A}\\mathbf{x}$$\n",
    "\n",
    "nuestro trabajo ahora es seleccionar $K$ de tal forma que los valores propios de la matriz $\\hat{A}$ den por lo menos un sistema estable. \n",
    "\n",
    "$$Re(\\lambda)<0 \\forall\\lambda \\in \\texttt{eig}(A-BKC)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### La matriz de estado del sistema en lazo cerrado\n",
    "\n",
    "Remplazamos los valores de las matrices y de $K=1$:\n",
    "\n",
    "$$\\hat{A}=\\left(A-BKC\\right)=\\left(\\left[\\begin{array}{}0&1\\\\0&0\\end{array}\\right]-\\left[\\begin{array}{}0\\\\1\\end{array}\\right]1\\left[\\begin{array}{}1&0\\end{array}\\right]\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\-1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0   1⎤\n",
       "⎢     ⎥\n",
       "⎣-1  0⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mA = sympy.Matrix([[0,1],[0,0]])-sympy.Matrix([0,1])*1*sympy.Matrix([[1,0]])\n",
    "display(mA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Analizando los valores propios del sistema tenemos que: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ - i : 1, \\  i : 1\\right\\}$"
      ],
      "text/plain": [
       "{-ⅈ: 1, ⅈ: 1}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(mA.eigenvals())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "El sistema es criticamente estable. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Respuesta de la particula al lazo cerrado\n",
    "\n",
    "Con el controlador propuesto, la particula se comporta como se muestra. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAAD4CAYAAADy46FuAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAW+UlEQVR4nO3dfZBd9X3f8fenIOQRuLGwBFZ4MNCqtvG4g+31BnAndTHYwHQQaZwYOh3LKYzq1rR2PQnIQ8fOkGSKyQwwmZDGiiHGjceASV0rjSgGAfUf5kELlREPwZJlt2ikQQrYjl05QoJv/7hHzfWy9+yu7tm9u837NXPnnvN7uPero5U+ex7uPakqJEka5O+MugBJ0sJmUEiSWhkUkqRWBoUkqZVBIUlqdfSoCzgSK1asqNNOO23UZUjSovL444//ZVWtnO28RRkUp512GhMTE6MuQ5IWlST/60jmeehJktTKoJAktTIoJEmtDApJUiuDQpLUqpOgSHJbkr1JnhrQnyS/l2RHkieTvKuvb22S7c1jbRf1SJK609UexReBC1v6LwJWN491wH8CSHI88FngF4Bx4LNJlndUkySpA50ERVV9E3ipZcga4EvV8wjwhiSrgA8C91XVS1X1A+A+2gNHWrg++cneQ/r/zHx94O4k4Pm+9V1N26D210iyjt7eCKeeeurcVCkNY+vWUVcgzYn5OpmdKdqqpf21jVUbqmqsqsZWrpz1J9AlSUdovoJiF3BK3/rJwO6WdknSAjFfQbER+Ehz9dPZwI+qag9wL/CBJMubk9gfaNokSQtEJ+coknwFeB+wIskuelcyLQGoqj8ENgEXAzuA/cCvNX0vJfktYEvzUtdVVdtJcUnSPOskKKrq8mn6C/j4gL7bgNu6qEOS1D0/mS1JamVQSJJaGRSSpFYGhSSplUEhSWplUEiSWhkUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgaFJKmVQSFJamVQSJJaGRSSpFYGhSSpVSdBkeTCJM8l2ZFk/RT9NyXZ2jy+k+SHfX2v9PVt7KIeSVJ3hr7DXZKjgFuAC4BdwJYkG6vqmcNjqurf943/t8A7+17ip1V11rB1SJLmRhd7FOPAjqraWVUvA3cAa1rGXw58pYP3lSTNgy6C4iTg+b71XU3bayR5M3A68EBf8+uSTCR5JMmlg94kybpm3MS+ffs6KFuSNBNdBEWmaKsBYy8D7q6qV/raTq2qMeCfAzcn+XtTTayqDVU1VlVjK1euHK5iSdKMdREUu4BT+tZPBnYPGHsZkw47VdXu5nkn8BA/e/5CkjRiXQTFFmB1ktOTHEMvDF5z9VKStwDLgYf72pYnWdosrwDeCzwzea4kaXSGvuqpqg4luQq4FzgKuK2qnk5yHTBRVYdD43LgjqrqPyz1NuDzSV6lF1rX918tJUkavaGDAqCqNgGbJrV9ZtL6b04x71vAO7qoQZI0N/xktiSplUEhSWplUEiSWhkUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgaFJKmVQSFJamVQSJJaGRSSpFYGhSSplUEhSWplUEiSWhkUkqRWnQRFkguTPJdkR5L1U/R/NMm+JFubx5V9fWuTbG8ea7uoR5LUnaHvcJfkKOAW4AJgF7AlycYpbml6Z1VdNWnu8cBngTGggMebuT8Yti5JUje62KMYB3ZU1c6qehm4A1gzw7kfBO6rqpeacLgPuLCDmiRJHekiKE4Cnu9b39W0TfbLSZ5McneSU2Y5lyTrkkwkmdi3b18HZUuSZqKLoMgUbTVp/c+A06rqHwL3A7fPYm6vsWpDVY1V1djKlSuPuFhJ0ux0ERS7gFP61k8GdvcPqKoXq+pAs/pHwLtnOleSNFpdBMUWYHWS05McA1wGbOwfkGRV3+olwLPN8r3AB5IsT7Ic+EDTJklaIIa+6qmqDiW5it5/8EcBt1XV00muAyaqaiPw75JcAhwCXgI+2sx9Kclv0QsbgOuq6qVha5IkdSdVU54SWNDGxsZqYmJi1GVIP+t97+s9P/TQKKuQBkryeFWNzXaen8yWJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrQwKSVKrToIiyYVJnkuyI8n6Kfo/leSZJE8m2ZzkzX19ryTZ2jw2Tp4rSRqtoe9wl+Qo4BbgAnr3wN6SZGNVPdM37H8CY1W1P8m/Bm4APtz0/bSqzhq2DknS3Ohij2Ic2FFVO6vqZeAOYE3/gKp6sKr2N6uPACd38L6SpHnQRVCcBDzft76raRvkCuCevvXXJZlI8kiSSwdNSrKuGTexb9++4SqWJM3Y0IeegEzRNuWNuJP8C2AM+Md9zadW1e4kZwAPJNlWVd99zQtWbQA2QO+e2cOXLUmaiS72KHYBp/StnwzsnjwoyfnAtcAlVXXgcHtV7W6edwIPAe/soCZJUke6CIotwOokpyc5BrgM+Jmrl5K8E/g8vZDY29e+PMnSZnkF8F6g/yS4JGnEhj70VFWHklwF3AscBdxWVU8nuQ6YqKqNwO8CxwFfTQLwv6vqEuBtwOeTvEovtK6fdLWUJGnEujhHQVVtAjZNavtM3/L5A+Z9C3hHFzVIkuaGn8yWJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAktTIoJEmtDApJUqtOvutJ+tusqnj44Yd5bNcufvzKK7z+5psZHx/nnHPOofkSTGlRMyikI3Tw4EFuvfVWbrjhBvbu3cvB/fs5WMWSa65hyZIlnHDCCVx99dVcccUVLFmyZNTlSkcsVYvvZnFjY2M1MTEx6jL0t9hPfvITLrroIp544gn2798/cNyyZct497vfzaZNmzjuuOPmsULptZI8XlVjs53nOQpplg4ePMhFF13Eli1bWkMCYP/+/Tz22GNcfPHFHDx4cJ4qlLplUEizdOutt/LEE09w4MCB6QcDBw4c4PHHH+e2226b48qkudFJUCS5MMlzSXYkWT9F/9Ikdzb9jyY5ra/v0037c0k+2EU90lypKm644YZp9yQm279/PzfccAOL8VCvNHRQJDkKuAW4CDgTuDzJmZOGXQH8oKr+PnAT8Llm7pn07rH9duBC4A+a15MWpIcffpi9e/dOP3AKL7zwAg8//HDHFUlzr4urnsaBHVW1EyDJHcAaoP/e12uA32yW7wZ+P73rBtcAd1TVAeB7SXY0r9f6r2nnvv/Dhz/vPzjNv+9svoufHnj5iOb+9MDLfPzmu/gH27xkVotLF4eeTgKe71vf1bRNOaaqDgE/At44w7kAJFmXZCLJhCcFNSoH/3o/rx46dERzXz10iEMHZnfISloIutijmOrXo8kHYgeNmcncXmPVBmAD9C6PvfNfnTObGqVO3PzTR7nm3iW8/PLs9yqWLj2GK897O5/wZ1cjctfHjmxeF3sUu4BT+tZPBnYPGpPkaODngJdmOFdaMMbHx4/4w3NHH30073nPezquSJp7XQTFFmB1ktOTHEPv5PTGSWM2Amub5Q8BD1Tv8o+NwGXNVVGnA6uBxzqoSZoT55xzDieccMIRzT3xxBM55xz3JrT4DB0UzTmHq4B7gWeBu6rq6STXJbmkGXYr8MbmZPWngPXN3KeBu+id+P7vwMer6pVha5LmShKuvvpqli1bNqt5y5Yt4+qrr/a7n7Qo+RUe0iwdPHiQ8847jy1btszoQ3dLly5lfHyczZs3+51PGim/wkOaJ0uWLOGee+5hfHx82j2LZcuWMT4+zqZNmwwJLVoGhXQEjjvuODZv3syNN97IGWecwbHHHsvShNDbgzj22GM544wzuPHGG9m8ebNfCKhFzUNP0pAO349iy0c+0rsfxSc/yfj4OGeffbbnJLSgHOmhJ+9HIQ0pCeeeey7nnnxyr+ETnxhtQVLHPPQkSWplUEiSWhkUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgaFJKmVQSFJamVQSJJaGRSSpFYGhSSp1VBBkeT4JPcl2d48L59izFlJHk7ydJInk3y4r++LSb6XZGvzOGuYeiRJ3Rt2j2I9sLmqVgObm/XJ9gMfqaq3AxcCNyd5Q1//b1TVWc1j65D1SJI6NmxQrAFub5ZvBy6dPKCqvlNV25vl3cBeYOWQ7ytJmifDBsWJVbUHoHk+oW1wknHgGOC7fc2/0xySuinJ0pa565JMJJnYt2/fkGVLkmZq2qBIcn+Sp6Z4rJnNGyVZBfxn4Neq6tWm+dPAW4H3AMcD1wyaX1UbqmqsqsZWrnSHRJLmy7R3uKuq8wf1JXkhyaqq2tMEwd4B4/4u8OfAf6iqR/pee0+zeCDJHwO/PqvqJUlzbthDTxuBtc3yWuDrkwckOQb4GvClqvrqpL5VzXPond94ash6JEkdGzYorgcuSLIduKBZJ8lYki80Y34V+EXgo1NcBvvlJNuAbcAK4LeHrEeS1LFpDz21qaoXgfdP0T4BXNks/wnwJwPmnzfM+0uS5p6fzJYktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAktTIoJEmtDApJUquhgiLJ8UnuS7K9eV4+YNwrfTct2tjXfnqSR5v5dzZ3w5MkLSDD7lGsBzZX1Wpgc7M+lZ9W1VnN45K+9s8BNzXzfwBcMWQ9kqSODRsUa4Dbm+Xb6d33ekaa+2SfB9x9JPMlSfNj2KA4sar2ADTPJwwY97okE0keSXI4DN4I/LCqDjXru4CTBr1RknXNa0zs27dvyLIlSTM17T2zk9wPvGmKrmtn8T6nVtXuJGcADyTZBvzVFONq0AtU1QZgA8DY2NjAcZKkbk0bFFV1/qC+JC8kWVVVe5KsAvYOeI3dzfPOJA8B7wT+FHhDkqObvYqTgd1H8GeQJM2hYQ89bQTWNstrga9PHpBkeZKlzfIK4L3AM1VVwIPAh9rmS5JGa9iguB64IMl24IJmnSRjSb7QjHkbMJHk2/SC4fqqeqbpuwb4VJId9M5Z3DpkPZKkjk176KlNVb0IvH+K9gngymb5W8A7BszfCYwPU4MkaW75yWxJUiuDQpLUyqCQJLUyKCRJrQwKSVIrg0KS1MqgkCS1MigkSa0MCklSK4NCktTKoJAktTIoJEmtDApJUiuDQpLUyqCQJLUyKCRJrYYKiiTHJ7kvyfbmefkUY/5Jkq19j79OcmnT98Uk3+vrO2uYeiRJ3Rt2j2I9sLmqVgObm/WfUVUPVtVZVXUWcB6wH/hG35DfONxfVVuHrEeS1LFhg2INcHuzfDtw6TTjPwTcU1X7h3xfSdI8GTYoTqyqPQDN8wnTjL8M+Mqktt9J8mSSm5IsHTQxybokE0km9u3bN1zVkqQZmzYoktyf5KkpHmtm80ZJVgHvAO7ta/408FbgPcDxwDWD5lfVhqoaq6qxlStXzuatJUlDOHq6AVV1/qC+JC8kWVVVe5og2NvyUr8KfK2qDva99p5m8UCSPwZ+fYZ1S5LmybCHnjYCa5vltcDXW8ZezqTDTk24kCT0zm88NWQ9kqSODRsU1wMXJNkOXNCsk2QsyRcOD0pyGnAK8D8mzf9ykm3ANmAF8NtD1iNJ6ti0h57aVNWLwPunaJ8Aruxb/z5w0hTjzhvm/SVJc89PZkuSWhkUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgaFJKmVQSFJamVQSJJaGRSSpFYGhSSplUEhSWplUEiSWhkUkqRWBoUkqZVBIUlqNVRQJPmVJE8neTXJWMu4C5M8l2RHkvV97acneTTJ9iR3JjlmmHokSd0bdo/iKeCfAd8cNCDJUcAtwEXAmcDlSc5suj8H3FRVq4EfAFcMWY8kqWNDBUVVPVtVz00zbBzYUVU7q+pl4A5gTZIA5wF3N+NuBy4dph5JUveGumf2DJ0EPN+3vgv4BeCNwA+r6lBf+2vuq31YknXAOoBTTz11biqVhnHWWaOuQJoT0wZFkvuBN03RdW1VfX0G75Ep2qqlfUpVtQHYADA2NjZwnDQyN9886gqkOTFtUFTV+UO+xy7glL71k4HdwF8Cb0hydLNXcbhdkrSAzMflsVuA1c0VTscAlwEbq6qAB4EPNePWAjPZQ5EkzaNhL4/9pSS7gHOAP09yb9P+80k2ATR7C1cB9wLPAndV1dPNS1wDfCrJDnrnLG4dph5JUvfS+8V+cRkbG6uJiYlRlyFJi0qSx6tq4GfeBvGT2ZKkVgaFJKmVQSFJamVQSJJaLcqT2Ul+DEz31SELwQp6nxdZ6BZDnYuhRrDOrllnt95SVa+f7aT5+AqPufDckZy5n29JJqyzG4uhRrDOrllnt5Ic0eWiHnqSJLUyKCRJrRZrUGwYdQEzZJ3dWQw1gnV2zTq7dUR1LsqT2ZKk+bNY9ygkSfPEoJAktVoUQZHkd5P8RZInk3wtyRsGjLswyXNJdiRZP4I6fyXJ00leTTLwUrkk30+yLcnWI71cbRizqHNk2zPJ8UnuS7K9eV4+YNwrzXbcmmTjPNbXum2SLE1yZ9P/aJLT5qu2SXVMV+dHk+zr24ZXjqDG25LsTfLUgP4k+b3mz/BkknfNd41NHdPV+b4kP+rblp8ZQY2nJHkwybPNv/FPTDFm9tuzqhb8A/gAcHSz/Dngc1OMOQr4LnAGcAzwbeDMea7zbcBbgIeAsZZx3wdWjHB7TlvnqLcncAOwvlleP9XfedP3kxFsv2m3DfBvgD9sli8D7lygdX4U+P35rm1SDb8IvAt4akD/xcA99O6KeTbw6AKt833AfxvxtlwFvKtZfj3wnSn+zme9PRfFHkVVfaP+5t7aj9C7G95k48COqtpZVS8DdwBr5qtGgKp6tqoW/CfGZ1jnqLfnGuD2Zvl24NJ5fO/pzGTb9Nd/N/D+JFPd/ncujfrvcEaq6pvASy1D1gBfqp5H6N0Zc9X8VPc3ZlDnyFXVnqp6oln+Mb17AJ00adist+eiCIpJ/iW9NJzsJOD5vvVdvHYDLRQFfCPJ40nWjbqYAUa9PU+sqj3Q++EHThgw7nVJJpI8kmS+wmQm2+b/jWl+yfkRvZtzzaeZ/h3+cnMI4u4kp0zRP2qj/lmcjXOSfDvJPUnePspCmsOd7wQendQ16+25YL7CI8n9wJum6Lq2qr7ejLkWOAR8eaqXmKKt82t/Z1LnDLy3qnYnOQG4L8lfNL+tdKaDOud8e7bVOIuXObXZlmcADyTZVlXf7abCgWaybebl53EaM6nhz4CvVNWBJB+jtxd03pxXNjsLYVvOxBPAm6vqJ0kuBv4rsHoUhSQ5DvhT4JNV9VeTu6eY0ro9F0xQVNX5bf1J1gL/FHh/NQfaJtkF9P82dDKwu7sKe6arc4avsbt53pvka/QOEXQaFB3UOefbs63GJC8kWVVVe5rd4r0DXuPwttyZ5CF6v0HNdVDMZNscHrMrydHAzzH/hy2mrbOqXuxb/SN65wAXmnn5tz2s/v+Qq2pTkj9IsqKq5vXLApMsoRcSX66q/zLFkFlvz0Vx6CnJhfTur31JVe0fMGwLsDrJ6UmOoXcCcd6ugpmpJMcmef3hZXon6qe8imLERr09NwJrm+W1wGv2gpIsT7K0WV4BvBd4Zh5qm8m26a//Q8ADA37BmUvT1jnp2PQl9I5pLzQbgY80V+ucDfzo8GHJhSTJmw6fh0oyTu//1xfbZ3VeQ4BbgWer6sYBw2a/PUd5hn4WZ/J30DumtrV5HL6a5OeBTZPO5n+H3m+U146gzl+il9YHgBeAeyfXSe8KlG83j6cXap2j3p70judvBrY3z8c37WPAF5rlc4FtzbbcBlwxj/W9ZtsA19H7ZQbgdcBXm5/dx4Az5vvveYZ1/sfm5/DbwIPAW0dQ41eAPcDB5ufyCuBjwMea/gC3NH+GbbRcUTjiOq/q25aPAOeOoMZ/RO8w0pN9/19ePOz29Cs8JEmtFsWhJ0nS6BgUkqRWBoUkqZVBIUlqZVBIkloZFJKkVgaFJKnV/wWZCWIB2KVsUAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.linspace(0,2*numpy.pi,40)\n",
    "x = numpy.sin(t)\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.plot([-2,2],[0,0])\n",
    "ax.plot([0,0],[-1,1],color='r')\n",
    "plt.xlim(-2,2);\n",
    "l, = ax.plot([0],[0],color='k',marker='.',markersize=30)\n",
    "\n",
    "animate = lambda i: l.set_data([x[i]], [0]);\n",
    "\n",
    "ani = FuncAnimation(fig, animate, frames=len(t), interval=1000/40);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x1c45cc90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(ani)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "**¿Por qué no se queda en el origen si este es el objetivo?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- No tenemos en cuenta la velocidad. \n",
    "- Necesitamos la información del estado (posición,velocidad) del sistema para estabilizarlo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Estabilizando la particula \n",
    "\n",
    "Para estabilizar la particula necesitamos conocer la información de todos los estados del sistema. Salvo que en nuestro sistema solo tenemos un sensor de posición:\n",
    "\n",
    "$$\\mathbf{y}=\\left[\\begin{array}{}1&0\\end{array}\\right]\\mathbf{x}$$\n",
    "\n",
    "El estado desconocido es la velocidad, el cual puede ser estimado de la posición. Por ahora supongamos que podemos medir ambos estados. \n",
    "\n",
    "$$\\mathbf{y}_{supuesto}=\\left[\\begin{array}{}1&0\\\\0&1\\end{array}\\right]\\mathbf{x}$$\n",
    "\n",
    "con esta nueva matriz $C$, proponemos un controlador $K$:\n",
    "\n",
    "$$K = \\left[\\begin{array}{}k_1&k_2\\end{array}\\right]$$\n",
    "\n",
    "Encontremos la nueva matriz $\\hat{A}$ para el sistema en lazo cerrado."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Nueva matriz de estado del sistema en lazo cerrado\n",
    "\n",
    "Recordemos que aquí estamos suponiendo que podemos medir ambos estados del sistema (posición y velocidad). Remplazamos los valores de las matrices y de $K$:\n",
    "\n",
    "$$\\hat{A}=\\left(A-BKC\\right)=\\left(\\left[\\begin{array}{}0&1\\\\0&0\\end{array}\\right]-\\left[\\begin{array}{}0\\\\1\\end{array}\\right]\\left[\\begin{array}{}k_1&k_2\\end{array}\\right]\\left[\\begin{array}{}1&0\\\\0&1\\end{array}\\right]\\right)$$\n",
    "\n",
    "Luego $\\hat{A}=$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\- k_{1} & - k_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 0    1 ⎤\n",
       "⎢        ⎥\n",
       "⎣-k₁  -k₂⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "k1,k2 = sympy.symbols('k_1 k_2')\n",
    "mA3 = sympy.Matrix([[0,1],[0,0]])-sympy.Matrix([0,1])*sympy.Matrix([[k1,k2]])*sympy.Matrix([[1,0],[0,1]])\n",
    "display(mA3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Los valores propios o polos del sistema son:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ - \\frac{k_{2}}{2} - \\frac{\\sqrt{- 4 k_{1} + k_{2}^{2}}}{2} : 1, \\  - \\frac{k_{2}}{2} + \\frac{\\sqrt{- 4 k_{1} + k_{2}^{2}}}{2} : 1\\right\\}$"
      ],
      "text/plain": [
       "⎧          _____________               _____________   ⎫\n",
       "⎪         ╱           2               ╱           2    ⎪\n",
       "⎨  k₂   ╲╱  -4⋅k₁ + k₂         k₂   ╲╱  -4⋅k₁ + k₂     ⎬\n",
       "⎪- ── - ────────────────: 1, - ── + ────────────────: 1⎪\n",
       "⎩  2           2               2           2           ⎭"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eigen = mA3.eigenvals()\n",
    "display(eigen)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Escogamos los valores del controlador"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "1306c8a5cc13423a9b4286e40ac38d7e",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "HBox(children=(FloatSlider(value=1.0, description='k1', max=10.0, orientation='vertical'), FloatSlider(value=2…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Parametros del modelo\n",
    "\n",
    "paramK1 = widgets.FloatSlider(value=1,min=0,max=10,step=0.1,description='k1', orientation=\"vertical\")\n",
    "paramK2 = widgets.FloatSlider(value=2,min=0,max=5,step=0.1,description='k2', orientation=\"vertical\")\n",
    "\n",
    "## Definicion de la simulacion\n",
    "\n",
    "def polos(K1,K2):\n",
    "    P1 = -K2/2+sympy.sqrt(K2**2-4*K1)/2\n",
    "    P2 = -K2/2-sympy.sqrt(K2**2-4*K1)/2\n",
    "    plt.scatter([sympy.re(P1),sympy.re(P2)],[sympy.im(P1),sympy.im(P2)])\n",
    "    plt.grid()\n",
    "    plt.title('Polos de la Particula en lazo cerrado con sensores de posición y velocidad')\n",
    "    plt.xlabel('Real')\n",
    "    plt.ylabel('Imaginario')\n",
    "    plt.ylim(-3,3)\n",
    "    plt.xlim(-6,0)\n",
    "\n",
    "## Presentación de los resultados    \n",
    "    \n",
    "plot_exponencial = widgets.interactive_output(polos,{'K1':paramK1,'K2':paramK2})      \n",
    "widgets.HBox([paramK1,paramK2,plot_exponencial])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "polos(3,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Verifiquemos el comportamiento de la particula\n",
    "\n",
    "- Tomemos los valores para el controlador $k_1=1$ y $k_2=1$\n",
    "- Tomemos los valores para el controlador $k_1=0.1$ y $k_2=1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Verifiquemos el comportamiento de la particula\n",
    "\n",
    "Tomemos los valores para el controlador $k_1=1$ y $k_2=2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.rcParams[\"animation.html\"] = \"html5\"#\"jshtml\"\n",
    "\n",
    "particula = control.ss('0 1;-1 -2','1;0','1 0','0')\n",
    "t,x = control.impulse_response(particula)\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.plot([-2,2],[0,0])\n",
    "ax.plot([0,0],[-1,1],color='r')\n",
    "plt.xlim(-2,2);\n",
    "l, = ax.plot([0],[0],color='k',marker='.',markersize=30)\n",
    "\n",
    "animate = lambda i: l.set_data([-x[i]], [0]);\n",
    "\n",
    "ani = FuncAnimation(fig, animate, frames=len(t), interval=1000/40);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0xdeec70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(ani)"
   ]
  },
  {
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    "# Los valores propios important\n",
    "\n",
    "- Es claro que algunos valores propios son mejores que otros. \n",
    "  - Algunos causan oscilaciones\n",
    "  - Algunos hacen que el sistema responda lentamente\n",
    "  - etc.\n",
    "\n",
    "- Próximamente veremos como selecionar los valores propios. "
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